Distributions Of Random Variables Research Articles

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∞.

Information entropy of the generalized beta distribution

The output distribution of statistical variables characterizes the level of uncertainty of a complex system, the state of which is given by probabilistic changes in internal relationships. Information entropy is an effective tool for analysing the distribution of data in complex systems. Subfamilies of generalized beta distributions of the first and second types includes many simple distributions as special cases. These subfamilies are often used to model the output state of the system by distributions of random variables. There are conditions in the paper, that are imposed on the parameters of the generalized beta distribution to obtain relationships with simpler types. The paper contains an expression for calculating the information entropy, the generalized beta distribution, which includes distributions of the first and second kind as special cases. Expressions of information entropies for the most famous subfamilies of generalized beta distribution also are given.

A Novel Tightly Coupled Solution for SINS/Polarized Navigation System/Odometer Integration Using Polarized and Installed Angle Errors Model

Precise and reliable autonomous navigation in a GPS-denied environment is critical to unmanned systems. The idea of combining SINS, the polarized navigation system (PNS), and the odometer (OD) inspired by desert ants has been proven to be effective for autonomous navigation. However, there are two major challenges for polarization navigation nowadays: inaccurate modeling and obtaining reliable heading information when some sensor channels are blocked. Aiming at these two problems, a tightly-coupled solution for SINS/PNS is proposed in this paper. To obtain a refined integrated navigation system model, the installation errors between the inertial units and PNS, and polarization angle calculation errors are analyzed and modeled for SINS/PNS. Then, to quickly gain the accurate state estimation, an improved iterative unscented filtering method is devised. In particular, the sigma-point updating step with the conditional distribution of high-dimensional Gaussian distribution random variables is developed, which employs partial states to sample in each iteration to reduce the calculation burden. Finally, a detection and elimination mechanism for the abnormal light channels is provided to enhance the reliability of the integration in the presence of a light blockage. The optical channels for navigation are chosen using this mechanism depending on the difference between the predicted and measured incident light intensities. The results in both simulations and outdoor experiments show that the proposed method provides a higher heading estimation accuracy than the traditional SINS/PNS navigation method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article is motivated by the inaccurate modeling and obtaining reliable heading in the presence of light occlusion for the polarization navigation. Various integrated navigation algorithms based on polarized skylight have been widely developed. However, the complex environment and inaccurate modeling limit the application of the polarization navigation. This article gives a novel accurate modeling method and reliable navigation algorithm. Tightly-coupled model with installation errors and the polarization angle error aims to improve the accuracy of the polarization navigation model. The improved iterative unscented Kalman filter is utilized to quickly obtain accurate state estimation. And the light detection and elimination mechanism is used to select normal light channels to navigate in the presence of light blockage. This navigation method can also extend the application of the polarization navigation.

Analysis on Probability Mass Function and Probability Density Function

Probability Mass Function (PMF) and Probability Density Function (PDF) are fundamental concepts in probability theory and statistics that play a crucial role in describing the probability distribution of random variables. This abstract provides a comprehensive overview of these concepts, highlighting their definitions, characteristics, and applications. The Probability Mass Function is a concept primarily associated with discrete random variables. It defines the probability of a specific outcome occurring. The PMF assigns probabilities to individual values in the sample space, providing a clear picture of the likelihood of each possible outcome. Commonly denoted as P(X=x), where X is the random variable and x is a specific value, the PMF must satisfy two essential properties: non-negativity and the sum of probabilities over all possible outcomes equals one. On the other hand, the Probability Density Function is a concept applied to continuous random variables. Unlike the PMF, which deals with specific values, the PDF deals with ranges of values. The PDF represents the probability that a continuous random variable falls within a given interval. Denoted as f(x), it is essential to note that the probability of any specific point is zero, and instead, probabilities are defined for intervals. The area under the PDF curve over a given interval corresponds to the probability of the random variable falling within that interval. Understanding the differences and similarities between PMF and PDF is crucial for statistical analysis. While PMF is discrete and deals with specific values, PDF is continuous and provides probabilities for intervals. Both functions are integral to the calculation of various statistical measures, including expected values, variance, and standard deviation. This abstract concludes with a discussion of practical applications in diverse fields, such as finance, engineering, and natural sciences, where a deep understanding of PMF and PDF is essential for making informed decisions and drawing meaningful conclusions from data. The integration of these concepts into statistical models and analyses enhances the accuracy and reliability of predictions, making PMF and PDF indispensable tools in the field of probability and statistics

A decreasing failure rate model with a novel approach to enhance the artificial neural network's structure for engineering and disease data analysis

The study focuses on key metrics used to examine the characteristics of a lifetime random variable distribution in reliability and survival theory research. In this analysis, metrics including the probability density function time, mean residual lifespan, mean time between failures, hazard rate, and reliability function are essential. The focus of the inquiry is these important parameters in relation to the Burr-Hatke exponential model specifically. The study focuses on key metrics used to examine the characteristics of a lifetime random variable distribution in reliability and survival theory research. In this analysis, metrics including the probability density function time, mean residual lifespan, mean time between failures, hazard rate, and reliability function are essential The focus of the inquiry is these important parameters in relation to the Burr-Hatke exponential model specifically. A key component of the research is a comparison of the outcomes from the artificial intelligence approach and those from conventional literature-based methodologies. This comparison study sheds light on how well the artificial neural network framework performs while evaluating the Burr-Hatke exponential model's technical features. The study allows a comprehensive analysis of the training and prediction capabilities of the growing neural network by calculating multiple performance measures. This comprehensive strategy improves our comprehension of the model's survival traits and reliability, offering significant contributions to the larger field of study. The network structure's mean square error was estimated to be 5.19E-04, and its coefficient of determination value was 0.99987 for the first neural network model. For the second neural network model, the coefficient of determination value was 0.99999 and the mean square error value was 4.58E-06. The outcomes amply revealed the neural network structure's extraordinarily high prediction accuracy and the degree to which the prediction outputs agree with those of the Maximum Likelihood Estimation technique.

Law of Probability Distribution Function of The Module of Difference of Discrete Random Variables

The use of the absolute value symbol in theoretical studies often leads to difficulties in analytical operations. However, from a practical point of view, transformations based on absolute differences are the most visual and effective. The article discusses functional transformations, which are modules of the differences of discrete random variables. The concept of the first difference modulus is given. The laws of the probability distribution of random variables obtained as the modulus of the difference of two independent random variables that have the same discrete distribution (Bernoulli or geometric), as well as the distribution laws of new random variables that are the sums of the moduli of the first difference, have been found. The main numerical characteristics of new random variables are calculated.

Systematic framework for handling uncertainty in probabilistic failure analysis of corroded concretes

Due to ambiguous correlations between input random variables and multi-source uncertainties from the electrochemical model, significant differences between deterministic corrosion prediction models and actual measurements are often observed. A systematic framework of quantification of uncertainties is developed for structures with correlated random variables originated from multiple sources, which allows efficiently estimating the failure probability distribution of the steel corrosion over time considering the randomness of the cover depth, the surface chloride concentration, and the chloride diffusion coefficient. After partitioning correlated random variables into different groups based on their uncertain sources, the Morris one-step-at-a-time and Sobol model is established to rank with respect to the importance of each correlated random variable. Based on polynomial chaos expansions and genetic programming methods, a more condensed set of random variables is created to propagate parametric problems. The unknown probability distribution of the input random variables is formulated by the Markov chain Monte Carlo to realize rigorous uncertainty quantification of the structural reliability. The application of the systematic framework to a set of numerical examples of steel corrosion includes experimental validation and uncertainty quantification and propagation of environmental, material and geometric properties. The results show that the framework can be integrated with parametric electrochemical models to allow robustness and reliability of corrosion prediction.

A dynamical Gaussian, lognormal, and reverse lognormal Kalman filter

AbstractWe derive a generalization of the Kalman filter that allows for non‐Gaussian background and observation errors. The Gaussian assumption is replaced by considering that the errors come from a mixed distribution of Gaussian, lognormal, and reverse lognormal random variables. We detail the derivation for reverse lognormal errors and extend the results to mixed distributions, where the number of Gaussian, lognormal, and reverse lognormal state variables can change dynamically every analysis time. We test the dynamical mixed Kalman filter robustly on two different systems based on the Lorenz 1963 model, and demonstrate that non‐Gaussian techniques generally improve the analysis skill if the observations are sparse and uncertain, compared with the Gaussian Kalman filter.

Robust learning of parsimonious deep neural networks

We propose a simultaneous learning and pruning algorithm capable of identifying and eliminating irrelevant structures in a neural network during the early stages of training. Simultaneous learning and pruning presents serious challenges such as premature pruning of units that can lead to poor performance. Our method is capable of overcoming such challenges and is robust, i.e., it gives consistent pruning levels and prediction accuracy regardless of weight initialization or the size of the starting network. Thus, it allows for substantial computational cost savings during training, besides that of inference, and it can enable the training of very deep networks when transfer learning to obtain fully trained deep networks that can be pruned after training is not possible. Our approach is based on variational inference principles using Gaussian scale mixture priors on the neural network weights. The variational posterior distribution of Bernoulli random variables multiplying the units/filters is learned, similarly to adaptive dropout. We construct a novel hyper-prior distribution over the prior parameters to impose properties crucial for their optimal selection and the overall robustness of our algorithm. It is shown in the context of our algorithm that the parameters of the posterior distributions practically converge to either 0 or 1, establishing a deterministic final network. Convergence is proved analytically based on dynamical systems theory and from the theoretical results, practical pruning conditions are established. The proposed algorithm is evaluated on the MNIST, CIFAR-10 and ImageNet data sets and the commonly used fully connected, convolutional and residual architectures LeNet, VGG16 and ResNet. The simulations show that our method typically achieves better pruning levels while maintaining test-accuracy on par with state-of the-art methods for structured pruning in a manner robust with respect to network initialization and initial size.

Frequency-Control-Aware Probabilistic Load Flow: An Analytical Method

Probabilistic load flow (PLF) calculation, as a fundamental tool to analyze transmission system behavior, has been studied for decades. Despite a variety of available methods, existing PLF approaches rarely take system control into account. However, system control, as an automatic buffer between the fluctuations in random variables and the variations in system variables (e.g., nodal voltages), has a significant impact on the final PLF result. To consider control actions' influence, this paper proposes the first analytical PLF method for the transmission grid that takes into account primary and secondary frequency controls. This method is based on a high-precision linear power flow model, whose precision is even further improved in this paper by an original correction approach. This paper also proves that if the joint probability distribution (JPD) of random variables is expressed by a Gaussian mixture model (GMM), then the JPD of system variables is an infinite GMM. By leveraging this proposition, the proposed method generates the joint PLF of the whole system at the quasi-steady and steady states. The high accuracy and satisfactory efficiency of this method are verified on multiple test systems with different percentages of renewables.

Time to Absorption in Markov Chains as a Mixture Distribution of Hypo-Exponential Distributions

Abstract We given an elementary proof that in a Markov chain with absorbing states, and positive probability of absorption at some time t &gt; 0 {t&gt;0} , time to absorption follows a mixture distribution of hypo-exponential random variables. We use this fact to show that early approximations of such a distribution yield the length of the shortest path from an initial state to an absorbing state. Thus different Markov chains with the same distance of shortest paths can yield identical first order approximations. Our work is motivated by the classical Armitage and Doll model of carcinogenesis.

Application of the Monte-Carlo Method to Assess the Operational Reliability of a Household-Constructed Wetland with Vertical Flow: A Case Study in Poland

The objective of this study was to model the operation of a vertical-flow constructed wetland (VF-CW) for domestic wastewater, using Monte-Carlo simulations and selected probability distributions of various random variables. The analysis was based on collected wastewater quality data, including the values of the pollutant indicators BOD5 (biochemical oxygen demand), CODCr (chemical oxygen demand), and TSS (total suspended solids), in the 2017–2020 period. Anderson–Darling (A–D) statistics were applied to assess the fit of the theoretical distributions to the empirical distributions of the random variables under study. The selection of the best-fitting statistical distributions was determined using the percentage deviation (PBIAS) criterion. Based on the analyses that were performed, the best-fitting statistical distributions for the pollution indicators of the raw wastewater were the generalised extreme value distribution for BOD5, the Gaussian distribution for CODCr, and the log-normal distribution for TSS. For treated effluent, the log-normal distribution was the best fit for BOD5 and CODCr; the semi-normal distribution, for TSS. The new data generated using the Monte-Carlo method allowed the reliability of the VF-CW operation to be assessed by determining the reliability indices, i.e., the average efficiency of the removal of pollutants (η), the technological efficiency index (R), the reliability index (CR), and the risk index of the negative control of the sewage treatment plant operation (Re). The obtained results indicate that only in the case of CODCr, the analysed treatment facility may fail to meet the requirements related to the reduction of organic pollutants to the required level, which is evidenced by the values of the indicators CR = 1.10, R = 0.49, and η = 0.82. In addition, the risk index of the negative operation of the facility (Re) assumes a value of 1, which indicates that during the period of its operation, the VF-CW system will not operate with the required efficiency in relation to this indicator. The novelty of this work is the implementation of the indicated mathematical simulation methods for analysing the reliability of the operation of the domestic wastewater treatment facility.

REIN: Reliability Estimation via Importance sampling with Normalizing flows

We introduce a novel framework called REIN: Reliability Estimation by learning an Importance sampling (IS) distribution with Normalizing flows (NFs). The NFs learn probability space maps that transform the probability distribution of the input random variables into a quasi-optimal IS distribution. NFs stack together invertible neural networks to construct differentiable bijections with efficiently computed Jacobian determinants. The NF ‘pushes forward’ a realization from the input probability distribution into a realization from the IS distribution, with importance weights calculated using the change of variables formula. We also propose a loss function to learn a NF map that minimizes the reverse Kullback–Leibler divergence between the ‘pushforward’ distribution and a sequentially updated target distribution obtained by modifying the optimal IS distribution. We demonstrate REIN’s efficacy on a set of benchmark problems that feature very low failure rates, multiple failure modes and high dimensionality, while comparing against other variance reduction methods. We also consider two simple applications, the reliability analyses of a thirty-four story building and a cantilever tube, to demonstrate the applicability of REIN to practical problems of interest. As compared to other methods, REIN is shown to be useful for high-dimensional reliability estimation problems with very small failure probabilities.

Reliability evaluation of folding wing mechanism deployment performance based on improved active learning Kriging method

The folding wing mechanism is essential for the normal flight of an aircraft, and it is necessary to evaluate the reliability of its deployment performance to ensure the long-term stable operation of the aircraft. In this study, a deployment dynamic model of the folding wing mechanism was developed and solved, and the essential factors affecting the deployment performance of the folding wing were analysed. The reliability evaluation models of the folding wing deployment performance were then presented for constant and non-constant random variable distribution parameters. Finally, the proposed improved AK-IS method was used to calculate the reliability of the deployment performance under the above two conditions. Through validation and analysis of the four cases, the results show that the improved AK-IS method is more efficient than the original AK-IS method, with guaranteed computational accuracy. Compared with the Monte Carlo Simulation, the improved AK-IS method also shows higher efficiency and good robustness in solving engineering problems, such as the reliability evaluation of deployment performance; thus, the proposed method can support the reliability evaluation of similar deployment mechanisms.

Reliability assessment of civil structures with incomplete probability distribution information

The probability distribution of random variables is usually incomplete owing to a limited amount of data, making it a challenging problem for structural reliability assessments. This paper presents a novel reliability analysis method for civil structures with incomplete data. The probability distributions of the random variables were estimated using an improved Bayesian inference method. In the proposed inference method, the frequencies and mode shapes were selected as the observation data. The likelihood function is replaced by homogeneous chaos, and the computational efficiency of Bayesian inference is improved. The affine invariant ensemble algorithm was used to generate samplers for the posterior distribution. After estimating the probability distribution of the incomplete random variables, a reliability analysis was performed, and the failure probability of the structure was estimated. Numerical studies on a two-span, two-story reinforced concrete building and a five-span bridge are presented. Experimental studies were also performed on an eight-story steel frame structure. The results show that the probability distributions of random variables can be significantly updated using the proposed technique and modal properties. The proposed technique was more efficient than the Gibbs sampling method. The failure probabilities of the structures were estimated using the estimated probability distributions.