Random Variables Research Articles

Dose Reconstruction for Epidemiological Studies among Ukrainian Chernobyl Cleanup Workers.

The present paper provides an overview of the methods and summarizes the results of estimating radiation doses and their uncertainties for Ukrainian-American epidemiological studies among the Chernobyl (Chornobyl) cleanup workers. After the Chernobyl accident occurred on April 26, 1986, more than 300,000 Ukrainian cleanup workers took part between 1986 and 1990 in decontamination and recovery activities at the site of the Chernobyl Nuclear Power Plant. The U.S. National Cancer Institute in collaboration with the Ukrainian National Research Center for Radiation Medicine conducted several epidemiological studies in this population. An important part of these studies was the reconstruction of the study participants' radiation doses and the assessment of uncertainties in doses. A method called realistic analytical dose reconstruction with uncertainty estimation (RADRUE) was used to calculate the doses from external irradiation during cleanup missions, which was the main exposure pathway for most study participants. At the initial phase of the accident during the atmospheric releases of radioactivity from the destroyed reactor, the cleanup workers also received doses from inhalation of radionuclides. In addition, study participants received doses at their places of residence, especially those who lived in highly contaminated areas. The radiation doses estimated for 2,048 male cleanup workers included in the Ukrainian-American epidemiological studies varied widely: (i) bone-marrow doses from external irradiation in the case-control study of leukemia of 1,000 cleanup workers ranged from 3.7 × 10-5 mGy to 3.3 Gy (mean = 92 mGy); (ii) thyroid doses in the case-control study of thyroid cancer in 607 persons from all exposure pathways combined were from 0.15 mGy to 9.0 Gy (mean = 199 mGy); (iii) gonadal doses in 183 cleanup workers from all exposure pathways combined in the study of germline mutations in the offspring after parental irradiation (trio study) ranged from 0.58 mGy to 4.1 Gy (mean = 392 mGy); (iv) thyroid doses in the human factor uncertainties study among 47 persons were from 20 mGy to 2.1 Gy (mean = 295 mGy); and (v) lung doses in the study of germline genetic variants associated with host susceptibility to COVID-19 estimated for 211 cleanup workers were from 0.024 mGy to 2.5 Gy (mean = 249 mGy). Doses of female cleanup workers were much lower than those of male cleanup workers: the mean doses for female cleanup workers were 27 mGy for 34 women included in the trio study and 56 mGy for 48 women participated in the study of germline genetic variants associated with host susceptibility to COVID-19. Uncertainties in dose estimates included two components: (i) inherent uncertainties arising from the stochastic random variability of the parameters used in exposure assessment and from a lack of knowledge about the true values of the parameters; and (ii) human factor uncertainties due to poor memory recall resulting in incomplete, inaccurate, or missing responses during personal interviews with cleanup workers conducted long after exposure. This paper also discusses possible developments and improvements in the methods to assess the radiation doses and associated uncertainties for cleanup workers.

Self-correlated spatial random variables: From an auto- to a sui- model respecification

This paper marks the 50-year publication anniversary of Besag's seminal spatial auto- models paper. His classic article synthesizes generic autoregressive specifications (i.e., a response variable appears on both sides of a regression equation and/or probability function equal sign) for the following six popular random variables: normal, logistic (i.e., Bernoulli), binomial, Poisson, exponential, and gamma. Besag dismisses these last two while recognizing failures of both as well as the more scientifically critical counts-oriented auto-Poisson. His initially unsuccessful subsequent work first attempted to repair them (e.g., pseudo-likelihood estimation), and then successfully revise them within the context of mixed models, formulating a spatially structured random effects term that effectively and efficiently absorbs and accounts for spatial autocorrelation in geospatial data. One remaining weakness of all but the auto-normal is a need to resort to Markov chain Monte Carlo (MCMC) techniques for legitimate estimation purposes. Recently, Griffith succeeded in devising an innovative uniform distribution genre—sui-uniform random variables—that accommodates spatial autocorrelation, too. Its most appealing feature is that, by applying two powerful mathematical statistical theorems (i.e., the probability integral transform, and the quantile function), it redeems Besag's auto- model failures. This paper details conversion of Besag's initial six modified variates, exemplifying them with both simulation experiments and publicly accessible real-world georeferenced data. The principal outcome is valuable spatial statistical advancements, with special reference to Moran eigenvector spatial filtering.

Strong Convergence Properties for Weighted Sums of Extended Negatively Dependent Random Variables Under Sub-linear Expectations with Statistical Applications

Strong Convergence Properties for Weighted Sums of Extended Negatively Dependent Random Variables Under Sub-linear Expectations with Statistical Applications

Reliability Evaluation of Multi-State Solar Energy Generating System with Inverters Considering Common Cause Failures

To ensure the efficient functioning of solar energy generation systems, it is crucial to have dependable designs and regular maintenance. However, when these systems or their components operate at multiple working levels, optimizing reliability becomes a complex task for models and analyses. In the context of reliability modeling in solar energy generation systems, researchers often assume that random variables follow an exponential distribution (binary-state representation) as a simplification, although this may not always hold true for real-world engineering systems. In the present paper, a multi-state solar energy generating system with inverters in series configuration is investigated, in which unreliable by-pass changeover switches, common cause failures (CCFs), and multiple repairman vacations are also considered. Furthermore, the arrivals of CCFs and the repair processes of the failed system due to CCFs are governed by different Markovian arrival processes (MAPs), and the lifetimes and repair times of inverters and by-pass changeover switches and the repairman vacation time in the system have different phase-type (PH) distributions. Therefore, the behavior of the system is represented using a Markov process methodology, and reliability measures for the proposed system are derived utilizing aggregated stochastic process theory. Finally, a numerical example and a comparison analysis are presented to demonstrate the findings.

Shaping of the risk of a series-parallel manufacturing structure maintenance according to quasi-coherence and the [formula omitted]-th survival value

The paper presents the authors’ model for determining the probability value of the survival function of the difference of two random variables: TP – the amount of time of correct operation and TB – the amount of time of failure. For this purpose, the concept of K-th survival value is introduced. The K-th survival value defines the probability that the total time of correct operation of the production system will be longer by at least k time units (for k≥0) than the total amount of time of failures. The model was exemplified by a real production system with a series–parallel structure. In the nesting manufacturing structure, the availability of machines determines the number and the type of flow streams of the processed material. To formalise the chains of relations at a given Δt, this paper utilises the quasi-coherence property. The paper identifies four possible cases of system operation: 1) a fully operational system in which every machine is capable of meeting the assumed production schedule; 2) a system which has lost the property of quasi-coherence (certain system objects are in an inoperable state) but it is possible to meet the assumed production schedule; 3) a system which has lost the property of quasi-coherence but the assumed task schedule is possible to meet after redefining the material flow streams; 4) a system which has lost the property of quasi-coherence and it is impossible to meet the assumed production schedule. Based on the definition of the survival function [1] S of the random variable T: ST:=PT>τ=1-FTτ,τ∈R, the concept of K-th survival value was introduced. For possible cases, K-th survival value was defined as: KT1,Ti=PT1-Ti≥k=1-FT1-Tik (for i=2,3,4), where T1 and Ti are random variables with given probability distributions. In turn, k is a deterministic value that determines the total number of time units of correct system operation in relation to the total failure time. The application of the developed algorithm enables the system to be analysed in terms of three aspects: (1) to determine the probability of on-time execution of tasks, taking into account historical values of machine failure and correct operation durations; (2) to determine the stability of the system based on empirical data; (3) to monitor the level of risk in real time. The paper validates the model by determining the stability of the analysed system for real machine failure data. The introduced criterion of stochastic stability of the production system has been verified for data in which distributions of correct operation times and failure times are random variables from the family of Gamma distributions. The presented model can be included in the group of multifactorial risk assessment methods.

On complete convergence for weighted sums of m-widely acceptable random variables under sub-linear expectations and its statistical applications

. In this article, by means of Rosenthal-type inequality, we study the complete convergence and strong law of large numbers for weighted sums of m-widely accpetable random variables under sub-linear expectations, which extend some existing results in classical probability space. As applications, we investigate the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations. In addition, we provide some simulation studies to verify the finite sample performance of the theoretical results.

Range expansions across landscapes with quenched noise

When biological populations expand into new territory, the evolutionary outcomes can be strongly influenced by genetic drift, the random fluctuations in allele frequencies. Meanwhile, spatial variability in the environment can also significantly influence the competition between subpopulations vying for space. Little is known about the interplay of these intrinsic and extrinsic sources of noise in population dynamics: When does environmental heterogeneity dominate over genetic drift or vice versa, and what distinguishes their population genetics signatures? Here, in the context of neutral evolution, we examine the interplay between a population's intrinsic, demographic noise and an extrinsic, quenched random noise provided by a heterogeneous environment. Using a multispecies Eden model, we simulate a population expanding over a landscape with random variations in local growth rates and measure how this variability affects genealogical tree structure, and thus genetic diversity. We find that, for strong heterogeneity, the genetic makeup of the expansion front is to a great extent predetermined by the set of fastest paths through the environment. The landscape-dependent statistics of these optimal paths then supersede those of the population's intrinsic noise as the main determinant of evolutionary dynamics. Remarkably, the statistics for coalescence of genealogical lineages, derived from those deterministic paths, strongly resemble the statistics emerging from demographic noise alone in uniform landscapes. This cautions interpretations of coalescence statistics and raises new challenges for inferring past population dynamics.

A novel efficient RFEM for reliability analysis and design of multi-line dynamically installed anchor for floating offshore wind turbines

A novel multi-line dynamically installed anchor was previously proposed by the authors to allow for the mooring of multiple floating offshore wind turbine, resulting in a significant reduction in the total number and cost of anchors required for floating wind farms. Considering the spatial variability of soil properties and the uncertainty of environmental loads, the present study performs reliability analysis and design of the multi-line dynamically installed anchor. Firstly, a strategy to repeatedly use fundamental random variables is proposed and validated for reducing the number of random variables used in Karhunen-Loève expansion in the simulation of random field of soil properties when the ratio of the soil domain dimension to the scale of fluctuation is large. Then, the efficiency, accuracy, and robustness of the RFEM (random finite element method) combined with K-MCS (Kriging model and Monte Carlo simulation) based on the proposed strategy are validated through examples of random capacity of foundations. Next, the random capacities and probabilistic VHMT failure envelopes of the multi-line dynamically installed anchor in spatially variable soil are investigated. Finally, the reliability design of multi-line dynamically installed anchors is conducted and compared with that of multi-line pile anchors, in which both the spatial variability of soil and the uncertainty of loads are condidered. The results show that the costs for multi-line dynamically installed anchors are obviously less than those of multi-line pile anchors.

Singularity properties of the entropy in an enclosed system characterized by accumulated noise stochastic processes

Due to the irreversibility of processes in an enclosed system, the randomness in the system increases in time and can be represented by an accumulated noise stochastic process. In contrast to the thermodynamics theory, the analysis of the system is based on the information theory point of view. The system is defined by two states: a time state and a timeless state. Based on the central limit theorem, due to the additivity of the noise process, the time state is characterized by the time-dependent Gaussian stochastic process. In contrast to Shannon's theory, precise expressions for the probability density, information, and entropy functions of the random variables defining the time-dependent process are derived and analyzed for a finite, infinite, and zero value of the related time-dependent variance. It has been proven that the entropy rate is not necessarily inversely proportional to time, as presented in previous works. Furthermore, mathematical proofs are presented, showing that the system entropy is a singularity function that increases towards infinity in the time state and then drops to zero value when the system enters the timeless state. The timeless state is characterized by the undefined pdf function, infinite values of information, and zero entropy.

Berry–Esseen-Type Estimates for Random Variables with a Sparse Dependency Graph

AbstractWe obtain Berry–Esseen-type bounds for the sum of random variables with a dependency graph and uniformly bounded moments of order $$\delta \in (2,\infty ]$$ δ ∈ ( 2 , ∞ ] using a Fourier transform approach. Our bounds improve the state-of-the-art obtained by Stein’s method in the regime where the degree of the dependency graph is large.

Adapted Chatterjee correlation coefficient

The need to accurately quantify dependence between random variables is a growing concern across various academic disciplines. Current correlation coefficients are typically intended for one of two purposes: testing independence or measuring relationship strength. Despite some attempts to address both aspects, the performance of these measures is still easily affected by oscillation and local noise. To address these limitations, we propose a new coefficient of correlation called the Adapted Chatterjee Correlation Coefficient (AC3). AC3 is designed to accurately identify both independence and functional dependence between variables, even in the presence of noise. We establish the consistency and asymptotic theories of (AC3). Additionally, we present a novel method, called Iterative Signal Detection Procedure (ISDP), for local signal identification. Our numerical studies and real data application demonstrate that AC3 outperforms state-of-the-art methods in terms of general performance and detecting local signals.

Response surface method-based efficient approach for the load and resistance factors calibration of breakwater foundation considering soil spatial variability

This study aims to propose an efficient approach to calibrate the load and resistance factors (LRFs) for the limit state design of breakwater foundations. The spatial variability of soil was considered by employing the Cholesky decomposition technique. To overcome the low computational efficiency of the Monte Carlo simulation (MCS), a response surface method (RSM) was utilized. Additionally, an efficient algorithm, namely the adaptive boundary-based particle filtering algorithm (ABB-PFA), was proposed with the use of RSM to automatically and quickly search optimal nominal values of random variables in the LRF calibration process. Noticeably, the optimal number of particles and values of the modification factor used in the ABB-PFA were observed to enhance the efficiency of the calibration process. The results showed that the computational efficiency of the calibration process is significantly increased by utilizing the response surface method and the proposed ABB-PFA. The efficiency increases the most if the number of particles is one and the value of the modification factor is 0.5 or 0.6. The influences of the random field element size on the efficiency of the calibration process were also investigated. By using the appropriate size, the efficiency is significantly increased while still obtaining a high accuracy of the LRFs calibration process.

Polynomial chaos expansion vs. Monte Carlo simulation in a stochastic analysis of wave propagation

The paper deals with the propagation of a stochastic wave in an elastic medium using the example of a seismic wave in a ground medium. Identification of subsoil parameters is never exact or complete which justifies the use of random field models or random variable models for input data; thus, the response of the subsoil is also random. In this paper and in the context of random variables, the focus is on a sensitivity analysis addressing the question of how the uncertainty of the input data (subgrade parameters) influences the obtained results (displacements). Two different methods of stochastic analysis are presented—the intrusive polynomial chaos approach supported by the Galerkin projection and Monte Carlo simulation—and compared by using an example of wave propagation in the elastic half-plane. Consistency in the results of both approaches has been achieved; however, the calculation efficiencies differ. The advantages and disadvantages of both approaches are discussed. The upper subsoil layer influences the variances of the random solutions much more than does the lower layer.

ANALYSIS OF CLUSTERING METHODS ON THE CAUSAL FACTORS OF DIABETES MELLITUS WITH FUZZY C MEANS METHOD

This study focuses on the effectiveness of clustering algorithms, namely Fuzzy C-Means by using k-Means algorithm as a supporting method, in the factors that cause Diabetes Mellitus. Diabetes mellitus is a chronic disease characterized by high levels of sugar (glucose) in the blood. Indonesia ranks 5th with the highest diabetes Mellitus patients in the world. This study aims to understand the pattern of factors causing Diabetes Mellitus and test the effectiveness of the clustering algorithm used. The data analysis methods include data collection, data pre-processing, distribution of cluster numbers, algorithm implementation, model adjustment, model training, model evaluation, and analysis of results. The results showed that the Fuzzy C-Means algorithm gets a coefficient of Fuzzynes score of 0.23 with a validation score of 0.40, while for supporting methods used K-Means algorithm gets a validation score of 0.32. This result shows that Fuzzy C-Means algorithm is superior in clastering the factors that cause Diabetes mellitus. The results of what variables have the most effect on cluster values 0 and 1. Where cluster 0 is a cluster that shows which variables are more at risk of diabetes, while cluster 1 is a cluster whose value shows what variables are far from the risk of causing diabetes mellitus. Then based on the results of the cluster that has been done, random blood sugar variables become the most influential variable on the risk of developing diabetes mellitus, followed by blood sugar variables 2 hours PP, and fasting blood sugar

Accurate estimation of the normalized mutual information of multidimensional data.

While the linear Pearson correlation coefficient represents a well-established normalized measure to quantify the inter-relation of two stochastic variables X and Y, it fails for multidimensional variables, such as Cartesian coordinates. Avoiding any assumption about the underlying data, the mutual information I(X, Y) does account for multidimensional correlations. However, unlike the normalized Pearson correlation, it has no upper bound (I ∈ [0, ∞)), i.e., it is not clear if say, I = 0.4 corresponds to a low or a high correlation. Moreover, the mutual information (MI) involves the estimation of high-dimensional probability densities (e.g., six-dimensional for Cartesian coordinates), which requires a k nearest-neighbor algorithm, such as the estimator by Kraskov et al. [Phys. Rev. E 69, 066138 (2004)]. As existing methods to normalize the MI cannot be used in connection with this estimator, a new approach is presented, which uses an entropy estimation method that is invariant under variable transformations. The algorithm is numerically efficient and does not require more effort than the calculation of the (un-normalized) MI. After validating the method by applying it to various toy models, the normalized MI between the Cα-coordinates of T4 lysozyme is considered and compared to a correlation analysis of inter-residue contacts.

Stability of equilibrium states for a stochastic difference equation of exponential form

It is well know that many process and problems in different fields of sciences and engineering can be modelled by linear and nonlinear difference equations. Particularly, in case of systems biology, ecology, biochemistry, genetics and physiology dynamics, many population models are governed by exponential difference equations, and lot of papers were published in this matter. The goal of this work is to study a nonlinear second order difference equation of exponential form: Xn+1 = bXn + cXne−σXn−1, n ≥ 0 where the parameters b, c have arbitrary values, σ > 0 and the initial values, X0, X−1 are arbitrary positive numbers. Our equation has an equilibrium points and is exposed to additive stochastic perturbations that are assumed a sequence of independent random variables with zero mean, unit variance and these perturbations are proportional to the deviation of the system state Xn from equilibrium points, and the result is stochastic difference equation. We cannot find the solutions of nonlinear stochastic difference equations of exponential form in all cases. So, one can study the behavior of solutions by asymptotic stability of equilibrium points. Additionally, with the general method of Lyapunov functional constructions for stochastic difference equations with discrete time, we provide the necessary and sufficient conditions for the asymptotic mean square stability of the two equilibrium points (zero and positive) within the linear approximation of the stochastic difference equation. These conditions also serve as sufficient conditions for the stability in probability of the equilibrium points of the initial nonlinear equation. Finally, to demonstrate the validity of the obtained results, some numerical examples and simulations of equations solutions are made and numerous graphical illustrations of stability trajectories of solutions are plotted.

On a preference relation between random variables related to an investment problem

On a preference relation between random variables related to an investment problem

Using joint probability density to create most informative unidimensional indices: a new method using pain and psychiatric severity as examples

BackgroundDimension reduction methods do not always reduce their underlying indicators to a single composite score. Furthermore, such methods are usually based on optimality criteria that require discarding some information. We suggest, under some conditions, to use the joint probability density function (joint pdf or JPD) of p-dimensional random variable (the p indicators), as an index or a composite score. It is proved that this index is more informative than any alternative composite score. In two examples, we compare the JPD index with some alternatives constructed from traditional methods.MethodsWe develop a probabilistic unsupervised dimension reduction method based on the probability density of multivariate data. We show that the conditional distribution of the variables given JPD is uniform, implying that the JPD is the most informative scalar summary under the most common notions of information. B. We show under some widely plausible conditions, JPD can be used as an index. To use JPD as an index, in addition to having a plausible interpretation, all the random variables should have approximately the same direction(unidirectionality) as the density values (codirectionality). We applied these ideas to two data sets: first, on the 7 Brief Pain Inventory Interference scale (BPI-I) items obtained from 8,889 US Veterans with chronic pain and, second, on a novel measure based on administrative data for 912 US Veterans. To estimate the JPD in both examples, among the available JPD estimation methods, we used its conditional specifications, identified a well-fitted parametric model for each factored conditional (regression) specification, and, by maximizing the corresponding likelihoods, estimated their parameters. Due to the non-uniqueness of conditional specification, the average of all estimated conditional specifications was used as the final estimate. Since a prevalent common use of indices is ranking, we used measures of monotone dependence [e.g., Spearman’s rank correlation (rho)] to assess the strength of unidirectionality and co-directionality. Finally, we cross-validate the JPD score against variance–covariance-based scores (factor scores in unidimensional models), and the “person’s parameter” estimates of (Generalized) Partial Credit and Graded Response IRT models. We used Pearson Divergence as a measure of information and Shannon’s entropy to compare uncertainties (informativeness) in these alternative scores.ResultsAn unsupervised dimension reduction was developed based on the joint probability density (JPD) of the multi-dimensional data. The JPD, under regularity conditions, may be used as an index. For the well-established Brief Pain Interference Inventory (BPI-I (the short form with 7 Items) and for a new mental health severity index (MoPSI) with 6 indicators, we estimated the JPD scoring. We compared, assuming unidimensionality, factor scores, Person’s scores of the Partial Credit model, the Generalized Partial Credit model, and the Graded Response model with JPD scoring. As expected, all scores’ rankings in both examples were monotonically dependent with various strengths. Shannon entropy was the smallest for JPD scores. Pearson Divergence of the estimated densities of different indices against uniform distribution was maximum for JPD scoring.ConclusionsAn unsupervised probabilistic dimension reduction is possible. When appropriate, the joint probability density function can be used as the most informative index. Model specification and estimation and steps to implement the scoring were demonstrated. As expected, when the required assumption in factor analysis and IRT models are satisfied, JPD scoring agrees with these established scores. However, when these assumptions are violated, JPD scores preserve all the information in the indicators with minimal assumption.

Stress–Strength Reliability Estimation for Gamma Distributions in a Noisy System

When assessing the reliability of a system or the viability of a material, it is necessary to consider the stress conditions of the operating system. Let X be the strength of the system and Y be the stress of the system caused by the environment. In a stress–strength model, the system’s reliability is defined as [Formula: see text], and due to ignoring environmental shocks or human failure, it is usually overestimated. To underestimate reliability, we assume that when X goes to the environmental conditions, the system’s strength reduces to [Formula: see text], where Z is a random variable on [Formula: see text]. We propose a modified reliability model with [Formula: see text] for the gamma distribution. In this model, stress Y and the scale mixture of the strength of the gamma distribution, [Formula: see text], are modeled as independent random variables. The proposed model is compared with the traditional model through a simulation study. Additionally, the asymptotic distribution is used to construct an asymptotic confidence interval for [Formula: see text]. Finally, we use two real datasets to demonstrate the practical applicability of our proposed method.

Resilient sampled-data control for fractional-order PMVG-based WTS with actuator saturation and probabilistic faults using fuzzy Lyapunov function method

The goal of this study is to present a new fuzzy Lyapunov function-based resilient sampled-data control method for a fractional-order permanent magnet vernier generator (PMVG)-based wind turbine system (WTS) that addresses actuator saturation and probabilistic faults. To achieve this, we first model the dynamical fractional model of the PMVG-based WTS as a T-S fuzzy model. Next, we introduce a fuzzy Lyapunov function within a unified framework that incorporates actuator saturation, probabilistic faults, and uncertainty information from control gain matrices. We then present an actuator fault model that represents actuator faults as stochastic variables with a specific probability distribution. Subsequently, we formulate a robust asymptotic stability criterion for closed-loop systems using linear matrix inequality (LMI), which integrates the fuzzy Lyapunov function and membership function information with fractional integral inequality techniques based on sampling time. We also propose an LMI approach to identify the domain of attraction that meets the necessary actuator saturation criteria. Finally, we apply the proposed method to a PMVG-based WTS, demonstrating its superiority and practical applicability through a comparison with existing methods using a numerical example of a nonlinear truck trailer system.