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The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [ABSTRACT FROM AUTHOR]

In this paper, we have considered the classical and Bayesian inference of the unknown parameters of the lifetime distribution based on the observations obtained from a simple step‐stress life‐testing (SSLT) experiment and when more than one cause of failures are observed. We have used the Tampered Random Variable (TRV) approach. The main advantage of the TRV approach is that it can be easily extended to a multiple step‐stress model as well as for different lifetime distributions. In this paper, it is assumed that the lifetime of the experimental units at each stress level follows Weibull distribution with the same shape parameter and different scale parameters. Further, we have introduced different tempering co‐efficient for different causes of failures. The maximum likelihood estimators and the associated asymptotic confidence intervals are obtained based on Type‐II censored observations. Further, we have considered the Bayesian inference of the unknown model parameters based on a fairly general class prior distributions. An extensive simulation study is performed to examine the performances of the proposed method, and the analysis of a real data set has been provided to show how the method can be used in practice. We have compared the TRV model with some of the other existing models, and the TRV model provides a better fit in terms of information theoretic criteria. We have also provided some optimality criteria, to determine the optimal stress change time and some sensitivity analyses have been performed. Most of the methods can be extended for other lifetime distributions also. [ABSTRACT FROM AUTHOR]

Considering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system. [ABSTRACT FROM AUTHOR]

In the postmarket drug and vaccine safety surveillance, when the number of adverse events follows a Poisson distribution, the ratio between the exposed and the unexposed person‐time information is the random variable that governs the decision rule about the safety of the drug or vaccine. The probability distribution function of such a ratio is derived in this paper. Exact point and interval estimators for the relative risk are discussed as well as statistical hypothesis testing. To the best of our knowledge, this is the first paper that provides an unbiased estimator for the relative risk based on the person‐time ratio. The applicability of this new distribution is illustrated through a real data analysis aimed to detect increased risk of occurrence of Myocarditis/Pericarditis following mRNA COVID‐19 vaccination in Manitoba, Canada. [ABSTRACT FROM AUTHOR]

This paper concerns the mixed H∞$$ {H}_{\infty } $$ and passivity‐based filtering problem for networked nonlinear systems with randomly occurring parameter uncertainties, multipath data packet dropouts, time‐varying delay, and quantization effects. The discrete‐time nonlinear plant considered in this paper is modeled as a Takagi–Sugeno fuzzy system with plant rules. The data missing phenomenon is considered in both measurement and performance output signals due to the unreliable nature of communication links. Further, to reduce the network bandwidth utilization, the measurement quantization is also employed for both the outputs before transmitting through the communication channel. Also, additive filter gain fluctuation is taken into account to reflect the imprecision in filter implementation. Stochastic variables obeying the Bernoulli distribution are incorporated in the model description to characterize the random occurrence of parameter uncertainties and data packet dropouts. A set of sufficient conditions assuring the stochastic stability of the filtering error system with mixed H∞$$ {H}_{\infty } $$ and passivity performance is derived according to Lyapunov stability theory. Finally, three numerical examples including mass–spring–damper and tunnel diode circuit model are established to establish the validity of the developed filter design algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes new algorithms of adaptive Gaussian filters for nonlinear state estimation with maximum one‐step randomly delayed measurements. The unknown random delay is modeled as a Bernoulli random variable with the latency probability known a priori. However, a contingent situation has been considered in this work when the measurement noise statistics remain partially unknown. Due to unavailability of the complete knowledge of measurement noise statistics, the unknown measurement noise covariance matrix is estimated along with states following: (i) variational Bayesian approach, (ii) maximum likelihood estimation. The adaptation algorithms are mathematically derived following both of the above approaches. Subsequently, a general framework for adaptive Gaussian filter is presented with which variants of adaptive nonlinear filters can be formulated using different rules of numerical approximation for Gaussian integrals. This paper presents a few of such filters, viz., adaptive cubature Kalman filter, adaptive cubature quadrature Kalman filter with their higher degree variants, adaptive unscented Kalman filter, and adaptive Gauss–Hermite filter, and demonstrates the comparative performance analysis with the help of a nontrivial Bearing only tracking problem in simulation. Additionally, the paper carries out relative performance comparison between maximum likelihood estimation and variational Bayesian approaches for adaptation using Monte Carlo simulation. The proposed algorithms are also validated with the help of an off‐line harmonics estimation problem with real data. [ABSTRACT FROM AUTHOR]

In this paper, we discuss about the hierarchical Bayesian (HB) estimation concerning system reliability for a waste‐to‐energy (WTE) process. The main goal of this approach is to obtain this estimation WTE process reliability under different loss functions related series, parallel, and k‐out‐of‐m systems. In this case, we can drive system outcome by estimating each individual component. It is assumed that components are independent and identically distributed exponential random variables. Properties of the HB estimations under different loss functions are also provided, and comparisons are made between Bayesian and HB estimators via Monte Carlo simulation. The implementation of the proposed procedure was illustrated in detail by employing numerical practical examples extracted from a WTE process at the final part of this paper. [ABSTRACT FROM AUTHOR]

Let X$X$ be a random variable and define its concentration function by Qh(X)=supx∈RP(X∈(x,x+h]).$$\begin{equation*} {\mathcal {Q}}_{h}(X)=\sup _{x\in {\mathbb {R}}}{\mathbb {P}}(X\in (x,x+h]). \end{equation*}$$For a sum Sn=X1+⋯+Xn$S_n=X_1+\cdots +X_n$ of independent real‐valued random variables, the Kolmogorov–Rogozin inequality states that Qh(Sn)⩽C∑i=1n(1−Qh(Xi))−12.$$\begin{equation*} {\mathcal {Q}}_{h}(S_n)\leqslant C{\left(\sum _{i=1}^{n}(1-{\mathcal {Q}}_{h}(X_i))\right)}^{-\frac{1}{2}}. \end{equation*}$$In this paper, we give an optimal bound for Qh(Sn)${\mathcal {Q}}_{h}(S_n)$ in terms of Qh(Xi)${\mathcal {Q}}_{h}(X_i)$, which settles a question posed by Leader and Radcliffe in 1994. Moreover, we show that the extremal distributions are mixtures of two uniform distributions each lying on an arithmetic progression. [ABSTRACT FROM AUTHOR]

The geometric process has been widely studied in various disciplines and applied in reliability, maintenance and warranty cost analysis, among others. In its applications in maintenance policy optimisation, the geometric process assumes constant repair effectiveness by its process rate. Nevertheless, in practice, maintenance effectiveness may differ from time to time and can therefore be better depicted by a random variable. Motivated by this argument, this paper proposes a new variant of the geometric process, which is referred to as the rate randomized geometric process (RRGP). The probabilistic properties of the RRGP are then investigated. The maximum likelihood method is utilised to estimate the parameters of the RRGP. Numerical examples are given to show its applicability in both maintenance policy optimization and fitting real‐world failure datasets. [ABSTRACT FROM AUTHOR]

Understanding and accurately characterizing energy dissipation mechanisms in civil structures during earthquakes is an important element of seismic assessment and design. The most commonly used model is attributed to Rayleigh. This paper proposes a systematic approach to quantify the uncertainty associated with Rayleigh's damping model. Bayesian calibration with embedded model error is employed to treat the coefficients of the Rayleigh model as random variables using modal damping ratios. Through a numerical example, we illustrate how this approach works and how the calibrated model can address modeling uncertainty associated with the Rayleigh damping model. [ABSTRACT FROM AUTHOR]

This paper proposes a probabilistic‐based framework to assess the failure probability of the existing shield tunnel owing to undercrossing tunnelling. A novel deterministic model using the two‐phase analysis method is presented to evaluate the longitudinal behaviours of the in‐service shield tunnel. First, the tunnelling‐induced settlement is estimated using the Loganathan and Poulos' method; second, the longitudinal beam‐spring model (LBSM), which can explicitly consider the circumferential joints, is applied to reflect the performances of the existing shield tunnel. Two well‐documented case histories are selected to validate the effectiveness of the deterministic analysis model. The predictions are also compared with previous analytical methods. Afterward, to improve the computational efficiency, a surrogate model is then established to replace deterministic model and perform the global sensitivity analysis (GSA). A probabilistic analysis is further performed to explore the tunnelling‐induced deformation characteristics of the shield tunnel by means of Monte Carlo simulation (MCS) method. Parametric analyses are further performed to explore the influences of key variables on the failure probabilities of the existing shield tunnel. The results show the proposed deterministic model gives a satisfactory prediction in tunnelling‐induced responses of the existing shield tunnel. The LBSM can reflect the segmental ring and joint deformations more realistically. The uncertainties in the ground elastic modulus, joint rotation stiffness and the joint shearing stiffness dominate the tunnel settlement, joint opening and dislocation, respectively. The failure probability of tunnel settlement is more sensitive to volume loss than that of the joint opening and dislocation. The whole system failure probability is identical to that of the tunnel settlement failure mode. By increasing the coefficient of variation (COV) of random variables, probability density function (PDF) curves for neutral and pessimistic conditions become lower and wider, and the maximum joint opening and dislocation positions corresponding to the peaks of PDF apparently deviate from that of the determinate results. Increasing the two tunnel clearances, the allowable volume loss for opening and dislocation increase linearly, but it has insignificant influence on the tunnel failure mode. However, enlarge the new tunnel diameter leads to rapid decrease in the allowable volume loss for all failure modes. [ABSTRACT FROM AUTHOR]

Aero‐disk is a key component of aero‐engine. Due to its complex working conditions, aero‐disk is prone to structural failures. Therefore, it is essential to analyze the reliability and importance input variables of aero‐disk. In this paper, a framework of aero‐disk reliability analysis and global sensitivity analysis (GSA) was established. First, the hazardous regions of aero‐disk were determined by finite element analysis, and the limit state function of aero‐disk was defined by the life interference model. Then D‐Vine model was utilized to establish the correlation model for aero‐disk hazardous regions to evaluate the reliability of aero‐disk. In addition, two GSA methods based on Copula and space partition were proposed to identify important input random variables considering underlying correlation, and the accuracy of the proposed method was verified by two examples. Finally, the proposed method was applied to the GSA of aero‐disk. The results show that the established framework fills in the gap of uncertainty analysis of aero‐disk, which can be extended to other engineering fields. The proposed GSA methods have both high efficiency and accuracy and can realize multi‐dimensional GSA when the correlation of input variables is considered. [ABSTRACT FROM AUTHOR]

The action responses of controlled mechanisms are often multiparametric, nonlinear, and uncertain. Complex dynamics and limited uncertain information pose difficulties for action reliability analysis. This paper develops an adaptive intelligent extremum surrogate model (AIESM) method for the action reliability under random‐interval hybrid uncertainty of a chain conveyor. First, a dynamic model of the chain conveyor is established, which considers the impact and frictional effects within the mechanical system, the regulating effects of the control system, and the external disturbances of the system. After that, a hybrid kernel extreme learning machine optimized by the sparrow search algorithm is employed as an intelligent surrogate model to construct the initial surrogate model from the hybrid uncertain variables to the limit state function (LSF) response and the extremum surrogate model (ESM) from the random variables to the LSF extremum response. An adaptive infilling strategy combining active learning and opposition‐based learning is applied to improve the accuracy and efficiency of the ESM and reduce the estimation error of action reliability. Finally, the action reliability interval bounds are obtained by Monte Carlo simulation based on the ESM. Two numerical examples are presented to illustrate the validity of the AIESM method. The action reliability interval of the chain conveyor provided by the proposed method is [0.9706, 0.9923]. [ABSTRACT FROM AUTHOR]

In this paper, an unscented Kalman filtering problem is studied for a nonlinear system with sensor saturation and randomly occurring false data injection attacks. A random variable obeying the Bernoulli distribution is employed to characterize the phenomena of the randomly occurring false data injection attacks. The aim of this paper is to design a modified unscented Kalman filter by minimizing an upper bound of filtering error covariance. Furthermore, a sufficient condition is provided to ensure an exponentially bounded filtering error in the mean square sense. Numerical simulations are presented to illustrate the validity of the proposed filter. [ABSTRACT FROM AUTHOR]

Most of the papers found in the literature concerned monitoring a process mean of a quality characteristic normally distributed. However, in several practical situations (like in reliability studies), the engineering team also desires to monitor a process mean in which the quality characteristic follows other probabilistic distributions, such as the Weibull. In this paper, we monitor a process mean of a random variable following a Weibull distribution using the traditional X¯$\overline{X}$ control chart. To improve the performance, Klein's two of two supplementary run rule is included in the X¯$\overline{X}$ chart. Such adoption promotes better performance, producing faster signals of abnormal situations (in terms of average run length) than the traditional X¯$\overline{X}$ chart in a great variety of shifts. An empirical example illustrates the proposed scheme. [ABSTRACT FROM AUTHOR]

Summary: This paper investigates a scalable recursive state estimation method for nonlinear two‐dimensional (2‐D) stochastic complex networks. In these networks, the switching topology is modelled by random variables generated by the Bernoulli distribution. Considering the Extended Kalman Filter structure, we employ the Taylor series expansion to handle the nonlinear functions in the model. Hence, high‐order terms emerging in the linearization process are modelled by norm‐bounded parameter uncertainties. Due to the uncertainty of such a system, we obtain an upper bound for the estimation error covariance. Furthermore, we obtain an upper bound for the cross covariance to reduce the proposed filter's computational complexity and improve scalability. Within the established setting, next, the proposed filter is designed in such a way that the presented upper bounds are minimized. Finally, a numerical example is provided to illustrate the effectiveness of the proposed state estimation algorithm. [ABSTRACT FROM AUTHOR]

A practical method is proposed in this paper for analysis of the lightning transient in the grounding grids. The circuit parameters of the grounding electrodes are evaluated by a set of analytic formulas, in which the soil ionization effect is taken into account. By means of the circuit parameters, the grounding electrode segments are represented by the π‐type circuits and a circuit model is built for the grounding grids. The lightning transient responses in the grounding grids can be calculated from the circuit model. The calculated results are compared with the measured ones to check the validity of the circuit model. In view of the actual distribution randomness, the lightning current amplitude is taken as a random variable and a statistical algorithm is further presented for statistical analysis of the lightning transient in the grounding grids. In the algorithm, the accumulative probability function is introduced to describe the random distribution of the lightning current amplitude. The lightning transient responses calculated in a large interval of the lightning current amplitude are weighted by the corresponding probability density. Then, the statistical values of the lightning transient responses can be obtained from the weighted sum. A numerical example is also given to discuss the difference between the statistical values and the nonstatistical ones calculated by the conventional method. © 2023 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

A new time‐dependent reliability analysis model with a mixture of random variables and interval variables is proposed in this paper. First, samples of random variables are generated according to their probability density functions. Then, the samples are substituted into the performance function to obtain the interval processes corresponding to these samples. The reliability index of each interval process is computed to finally obtain the upper and lower bounds of failure probability. The adaptive Kriging surrogate model based on the global addition strategy is used in the calculation process to increase the computational efficiency. The accuracy of the computational strategy is demonstrated by a mathematical example and three engineering practical cases. The proposed new hybrid time‐dependent reliability model is compared to the conventional hybrid reliability model. The comparison results show that the proposed model is more convenient to implement than the conventional reliability model. It can accurately evaluate the reliability of the structure and can be applied to real‐world engineering practice thereby providing assistance and guidance for hybrid time‐dependent reliability analysis of structures. [ABSTRACT FROM AUTHOR]

In steady state, the concentration n of a cosmogenic nuclide in a surface can be used to study the local erosion rate. When the erosion is due to episodic spalling events, this concentration is not constant in time but has a statistical distribution which reflects the underlying spallation process. In this paper, we study a model in which slabs of rock of depth w are removed in instantaneous events; the time intervals t between these events are random variables. The probability density for t, p(t), is taken to have a Pareto distribution, with a power law decrease at large values of t. This case is interesting to study, since the Pareto distribution can lead to the occurrence of a wide range of time intervals, which can lead to a broad distribution of measured values of n. We derive analytic expressions for the average and the variance of n in steady state and compare our results to previous work, which took p(t) to decrease exponentially with t. We also show results from simulations of the Pareto model, which allow us to gain insight into the time‐dependent behavior of the nuclide concentration. Key Points: We study a model for cosmogenic nuclide production in a surface undergoing episodic removal of slabs of materialThe time intervals between the events are random variables, with a power law probability densityWe derive exact expressions for the average and the variance of the nuclide concentration and discuss the implications [ABSTRACT FROM AUTHOR]

Summary: This paper deals with the filtering problem for a class of discrete‐time state‐saturated systems subject to randomly occurring nonlinearities and missing measurements. A set of mutually independent Bernoulli random variables is used to describe the random occurrence of the missing measurements. Due to the simultaneous consideration of the state saturation, the randomly occurring nonlinearities, and the missing measurements, it is extremely hard to calculate the actual filtering error covariance in a closed form. As such, the objective of this paper is to construct an upper bound for the filtering error covariance and then design the filter parameters to minimize such an upper bound. The performance of the proposed filters is analyzed in terms of boundedness and monotonicity. Specially, we have shown that the minimum upper bound is always bounded under a mild assumption. Moreover, the relationship between the estimator performance and the arrival probability of the measurements is discussed. A numerical simulation is used to demonstrate the effectiveness of the filtering method. [ABSTRACT FROM AUTHOR]

Isotonic regression or monotone function estimation is a problem of estimating function values under monotonicity constraints, which appears naturally in many scientific fields. This paper proposes a new Bayesian method with global–local shrinkage priors for estimating monotone function values. Specifically, we introduce half shrinkage priors for positive valued random variables and assign them for the first‐order differences of function values. We also develop fast and simple Gibbs sampling algorithms for full posterior analysis. By incorporating advanced shrinkage priors, the proposed method is adaptive to local abrupt changes or jumps in target functions. We show this adaptive property theoretically by proving that the posterior mean estimators are robust to large differences and that asymptotic risk for unchanged points can be improved. Finally, we demonstrate the proposed methods through simulations and applications to a real data set. [ABSTRACT FROM AUTHOR]

Recent technological advances have led to a rapid increase in the collection of capture–recapture data in continuous time. Unlike traditional capture–recapture data, the detection times from these technologies are themselves random variables and analysis of these data, therefore, requires models that properly account for stochasticity in both state transitions and detection times.Despite the ubiquity of continuously collected capture–recapture data, the mathematical concepts needed to fit continuous‐time models remain unfamiliar to many ecologists. In this paper, I provide an introduction to continuous‐time models, with a focus on multi‐state capture–recapture data.After reviewing the basic structure of these models, I describe several variations, including constant parameters, temporal variation in state transition rates and autocorrelation in detections. To aid comprehension, each model is accompanied by code to simulate data and fit the model in Stan.Although the models presented in this guide are only a small subset of the variations that are possible to suit the needs of specific datasets or questions, the concepts and code will hopefully serve as a foundation for future analyses, allowing ecologists to develop new and creative approaches to continuous‐time modelling. [ABSTRACT FROM AUTHOR]

Profile monitoring has received much attention from the applications in statistical process control. It is a method for monitoring the stability of a functional relationship between a response variable and one or more explanatory variables over the time axis. General linear profiles monitoring is very important since the linear relationship between a response variable and explanatory variables is easy to characterize besides it is simple and flexible. In addition, most of general linear profiles monitoring techniques assume normality of random error variables. However, the normality of random error variables is not satisfied in certain applications. This causes the existing monitoring methods for general linear profiles both inadequate and inefficient. Based on the log‐linear modelling, in this paper, we develop a non‐parametric control chart for Phase II monitoring of general linear profiles where normality of random error variables is not assumed. The proposed charting method applies the CUSUM (cumulative sum) to the Pearson chi‐square test for the Wilcoxon‐type rank‐based estimators of coefficient parameters and an estimator of random error variance. Effectiveness of the developed control chart is assessed and compared with that of two existing control charts based on the criterion of ARL (average run length). An industrial production process example is also applied to illustrate how the proposed control chart can be used in practice. [ABSTRACT FROM AUTHOR]

Summary: This paper concentrates on the problem of asynchronous ℋ∞$$ {\mathscr{H}}_{\infty } $$ filtering for singular Markov jump systems under DoS attacks, in which the unknown probability information cases are considered. To better describe cyber security problem, DoS attacks are considered as encountered randomly and modeled by a set of stochastic variables obeying Bernoulli distribution. Besides, hidden Markov model with partially known probability information is proposed to handle the cases that the Markov state information of the system is restrictedly accessed, and the transition probability information and observation probability information can not be directly obtained. A set of criteria are derived to analyze the regularity, causality, and stochastic stability for the filtering error system with an ℋ∞$$ {\mathscr{H}}_{\infty } $$ performance index. Furthermore, an asynchronous filter design method is proposed to against DoS attacks. Finally, a numerical example and a practical example modeled by a DC motor are given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Complex systems seen either in general engineering practice or economics are subjected to ever increased uncertainties that are mostly represented as random variables or parameters, and the characteristics of random variables are represented by their probability density functions (PDFs). Controlling their PDFs means to shape their stochastic distributions and in general it would provide a full treatment for system analysis and operational control and optimization. This leads to the development of stochastic distribution control (SDC) systems theory in the past decades, where the original aim of the controller design is to realize a shape control of the distributions of certain random variables in their PDFs sense for some engineering processes. Indeed, once the PDFs of these random variables or parameters are used to describe their distribution characters, the control task is to obtain control signals so that the output PDFs of stochastic systems are made to follow their target PDFs. The subject of SDC was initially originated for non‐Gaussian stochastic control systems design but has found a wide spectrum of applications in general systems in terms of data‐driven modeling, analysis, signal processing (filtering), data mining via multivariable statistics, decision‐making (optimization) for systems subjected to uncertainties and even in economics. In this context, SDC constitutes an effective primer tool for complex system analysis, control and operational optimizations. In this review paper, a detailed survey of the developments on the research of SDC systems will be made together with their wide spectrum applications and future perspectives. [ABSTRACT FROM AUTHOR]

The problem of an inspection permutation or inspection strategy (first discussed in a research paper in 1989 and reviewed in another research paper in 1991) is revisited. The problem deals with an N-component system whose times to failure are independent but not identically distributed random variables. Each of the failure times follows an exponential distribution. The components in the system are connected in series such that the failure of at least one component entails the failure of the system. Upon system failure, the components are inspected one after another in a hierarchical way (called an inspection permutation) until the component causing the system to fail is identified. The inspection of each component is a process that takes a non-negligible amount of time and is performed at a cost. Once the faulty component is identified, it is repaired at a cost, and the repair process takes some time. After the repair, the system is good as new and is put back in operation. The inspection permutation that results in the maximum long run average net income per unit of time (for the undiscounted case) or maximum total discounted net income per unit of time (for the discounted case) is called the optimal inspection permutation/strategy. A way of determining an optimal inspection permutation in an easier fashion, taking advantage of the improvements in computer software, is proffered. Mathematica is used to showcase how the method works with the aid of a numerical example. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

To fully benefit from remotely sensed observations of the terrestrial water cycle, bias and random errors in these data sets need to be quantified. This paper presents a Bayesian hierarchical model that fuses monthly water balance data and estimates the corresponding data errors and error‐corrected water balance components (precipitation, evaporation, river discharge, and water storage). The model combines monthly basin‐scale water balance constraints with probabilistic data error models for each water balance variable. Each data error model includes parameters that are in turn treated as unknown random variables to reflect uncertainty in the errors. Errors in precipitation and evaporation data are parameterized as a function of multiple data sources, while errors in GRACE storage observations are described by a noisy sine wave model with parameters controlling the phase, amplitude, and randomness of the sine wave. Error parameters and water balance variables are estimated using a combination of Markov Chain Monte Carlo sampling and iterative smoothing. Application to semiarid river basins in Iran yields (a) significant reductions in evaporation uncertainty during water‐stressed summers, (b) basin‐specific timing and amplitude corrections of the GRACE water storage dynamics, and (c) posterior water balance estimates with average standard errors of 4–12 mm/month for water storage, 3.5–7 mm/month for precipitation, 2–6 mm/month for evaporation, and 0–2 mm/month for river discharge. The approach is readily extended to other data sets and other (gauged) basins around the world, possibly using customized data error models. The resulting error‐filtered and bias‐corrected water balance estimates can be used to evaluate hydrological models. Key Points: A Bayesian hierarchical model fuses water balance data containing unknown bias and random errorsThe model is solved using a combination of Markov Chain Monte Carlo sampling and iterative smoothingComputed posteriors provide hydrologically consistent data error and water balance estimates [ABSTRACT FROM AUTHOR]

This paper presents the comparison of advanced semi‐probabilistic methods for the design and assessment of concrete structures represented by mathematical models solved by non‐linear finite element methods. The special attention is given to the advanced methods focused on the estimation of the coefficient of variation of structural resistance. Numerical examples represent a replication of laboratory experiments of beams with different failure modes. The obtained results are discussed with respect to the accuracy of the employed methods and the influence of the assumed statistical correlation among basic variables. Simplified methods give a good estimation of the design values, though their accuracy is dependent on the type of the failure mechanism. Moreover, it is shown that mutual correlations among random variables may significantly affect the design value of resistance, and they should be carefully defined and modeled. [ABSTRACT FROM AUTHOR]

In this paper, we discuss stochastic comparison of the largest order statistics arising from two sets of dependent distribution‐free random variables with respect to multivariate chain majorization, where the dependency structure can be defined by Archimedean copulas. When a distribution‐free model with possibly two parameter vectors has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we obtain the first sample maxima is larger than the second sample maxima with respect to the usual stochastic order, based on certain conditions. Applications of our results for scale proportional reverse hazards model, exponentiated gamma distribution, Gompertz–Makeham distribution, and location‐scale model, are also given. Meanwhile, we provide two numerical examples to illustrate the results established here. [ABSTRACT FROM AUTHOR]

This paper shows that in some axioms regarding the mixture of random variables, the requirement that the conclusions hold for all values of the mixture parameter can be weakened by requiring the existence of only one nontrivial value of the parameter, which need not be fixed. This is the case for the independence, betweenness, and mixture symmetry axioms. Unlike the standard axioms, these weaker versions cannot be refuted by experimental methods. [ABSTRACT FROM AUTHOR]

In this paper we used two sets of calculated molecular descriptors to predict blood-brain barrier (BBB) entry of a collection of 415 chemicals. The set of 579 descriptors were calculated by Schrodinger and TopoCluj software. Polly and Triplet software were used to calculate the second set of 198 descriptors. Following this, modelling and a two-deep, repeated external validation method was used for QSAR formulation. Results show that both sets of descriptors individually and their combination give models of reasonable prediction accuracy. We also uncover the effectiveness of a variable selection approach, by showing that for one of our descriptor sets, the top 5% predictors in terms of random forest variable importance are able to provide a better performing model than the model with all predictors. The top influential descriptors indicate important aspects of molecular structural features that govern BBB entry of chemicals. [ABSTRACT FROM AUTHOR]

Let (Zk)k⩾1$(Z_k)_{k\geqslant 1}$ be a sequence of independent and identically distributed complex random variables with common distribution μ$\mu$ and let Pn(X):=∏k=1n(X−Zk)$P_n(X):=\prod _{k=1}^n (X-Z_k)$ be the associated random polynomial in C[X]$\mathbb {C}[X]$. Kabluchko established the conjecture stated by Pemantle and Rivin that the empirical measure νn$\nu _n$ associated with the critical points of Pn$P_n$ converges weakly in probability to the base measure μ$\mu$. In this note, we establish that the convergence, in fact, holds in the almost sure sense. Our result positively answers a question raised by Kabluchko and formalized as a conjecture in the recent paper (Michelen and Vu [arXiv:2212.11867]). [ABSTRACT FROM AUTHOR]

System reliability directly affects the realization of software‐intensive system functions, which rely on both hardware and software. The operating status of hardware and software affect each other, and the continuous accumulation of their degraded performances will accelerate the occurrence of failures. However, the interaction relationship between hardware and software is often overlooked in most current research. In this paper, a new reliability analysis method is proposed for multi‐state software‐intensive systems with degradation accumulation effect and information transmission attenuation based on goal oriented (GO) methodology. First, a new function GO operator is created, and its operation formula is derived with the aid of Markov process theory. To quantify the impact of degradation accumulation on availability, non‐negative constant and random variable critical thresholds are set for the degradation state. Then, an effective optical power loss model is established to analyze the reliability of information transmission in the presence of attenuation. Next, the reliability analysis framework for multi‐state software‐intensive system is established based on the GO method. Finally, the proposed reliability analysis method is applied to the example of meter and decameter wavelength radio heliograph to demonstrate the applicability and feasibility of our work. The proposed model provides a more accurate reflection of the system reliability under actual operating conditions. [ABSTRACT FROM AUTHOR]

Joint models for longitudinal and survival data (JMLSs) are widely used to investigate the relationship between longitudinal and survival data in clinical trials in recent years. But, the existing studies mainly focus on independent survival data. In many clinical trials, survival data may be bivariately correlated. To this end, this paper proposes a novel JMLS accommodating multivariate longitudinal and bivariate correlated time‐to‐event data. Nonparametric marginal survival hazard functions are transformed to bivariate normal random variables. Bayesian penalized splines are employed to approximate unknown baseline hazard functions. Incorporating the Metropolis‐Hastings algorithm into the Gibbs sampler, we develop a Bayesian adaptive Lasso method to simultaneously estimate parameters and baseline hazard functions, and select important predictors in the considered JMLS. Simulation studies and an example taken from the International Breast Cancer Study Group are used to illustrate the proposed methodologies. [ABSTRACT FROM AUTHOR]

This paper is devoted to a new two‐parameter distribution on ℤ having an original additive stochastic structure. It is defined by the sum of two independent discrete random variables, where one is defined based on the Laplace distribution and the other on a transformed version of a distribution that can be viewed as a relative binomial distribution. It can thus also be viewed as a perturbed version of the Laplace distribution. We present some interesting properties of the proposed distribution, such as its moments (ordinary, non‐central, incomplete moments, etc.), characteristic function, skewness, and kurtosis, among others. The distribution is also applied to practical datasets from finance and health studies, where it is seen to perform better than some of the existing models defined on ℤ. [ABSTRACT FROM AUTHOR]

This paper employs stochastic analysis to investigate the combined effect of temporal and spatial variability on the temporal variance of baseflow in large catchments. The study makes use of the well‐known aggregated reservoir model, representing the catchment as a network of parallel linear reservoirs. Each reservoir models a sub‐catchment as an independent unit whose discharge temporal variation is characterized by a response time. By treating the rainfall‐generated recharge and the sub‐catchment response times as random variables, the statistical temporal moments of total baseflow are quantified. Comparisons are made between the temporal variance of baseflow in the aggregated reservoir model and that of a single homogeneous reservoir to define an upscaled response time. The analysis of the statistical moments of the random baseflow reveals that the number of reservoirs N has a weak impact on baseflow variance, with ergodic conditions achieved even with a small number of reservoirs. The study highlights that the ratio between the recharge correlation time and the geometric mean of the sub‐catchment response times plays a critical role in baseflow damping and the upscaled response. The results indicate that the dynamics of baseflow generation depend not only on the catchment hydro‐geological structure but also on the variability of the input signal. This research underscores the importance of understanding the combined influences of hydro‐geological factors and recharge input variability for baseflow prediction under uncertainty. The present study should be regarded as a first step, setting the theoretical framework for future research toward incorporating field data. Key Points: Stochastic model to investigate the effect of temporal and spatial variability on the baseflow statistics in large heterogeneous catchmentsBaseflow statistics depend strongly on the ratio of recharge timescale and the geometric mean of reservoir timescalesThe temporal variance of baseflow is used to define an equivalent reservoir with its characteristic timescale [ABSTRACT FROM AUTHOR]

δ$$ \delta $$‐shock model is one of the widely studied shock models in reliability theory and applied probability. In this model, the system fails due to the arrivals of two consecutive shocks which are too close to each other. That is, the system breaks down when the time between two successive shocks falls below a fixed threshold δ. In the literature, the δ‐shock model has been mostly studied by assuming that the time between shocks have continuous distribution. In the present paper, the discrete time version of the model is considered. In particular, a proper waiting time random variable is defined based on a sequence of two‐state Markov dependent binary trials and the problem of finding the distribution of the system's lifetime is linked with the distribution of the waiting time random variable, and we study the joint as well as the marginal distributions of the lifetime, the number of shocks and the number of failures associated with these binary trials. [ABSTRACT FROM AUTHOR]

The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener‐Itô integrals. [ABSTRACT FROM AUTHOR]

We consider a system of dependent Poisson variables, where each variable is the sum of an independent variate and a common variate. It is the common variate that creates the dependence. Within this system, a test of independence may be constructed where the null hypothesis is that the common variate is identically zero. In the present paper, we consider the maximum log likelihood ratio test. For this test, it is well‐known that the asymptotic distribution of the test statistic is an equal mixture of zero and a chi‐square distribution with one degree of freedom. We examine a Bartlett correction of the test, in the hope that we will get better approximation of the nominal size for moderately large sample sizes. A correction of this type is explicitly derived, and its usefulness is explored in a simulation study. For practical purposes, the correction is found to be useful in dimension two, but not in higher dimensions. [ABSTRACT FROM AUTHOR]

Correlations belong to the standard repertoire of ecologists for quantifying the strength of dependence between two random variables. Classical dependence measures are usually not capable of detecting non‐monotonic or non‐functional dependencies. Furthermore, they completely fail to detect asymmetry and direction in dependence, which exist in many situations and should not be ignored.In this paper, we present qad (short for quantification of asymmetric dependence), a nonparametric statistical method to quantify directed and asymmetric dependence of bivariate samples. Qad is applicable in general (e.g. linear, non‐linear, or non‐monotonic) situations, is sensitive to noise in data, exhibits a good small sample performance, detects asymmetry in dependence, shows high power in testing for independence, requires no assumptions regarding the underlying distribution of the data and reliably quantifies the information gain/predictability of quantity Y given knowledge of quantity X, and vice versa (i.e. q(X,Y) ≠q(Y,X)).Here, we briefly recall the methodology underlying qad, introduce the functions of the R‐package qad, which returns estimates for the measures qX,Y denoting the directed dependence of Y on X (or, equivalently, the influence of X on Y), qY,X the directed dependence of X on Y, aX,Y≔qX,Y−qY,X the asymmetry in dependence. Furthermore, qad can be used to predict Y given knowledge of X, and vice versa. Additionally, we compare empirical performance of qad with that of seven other well established measures and demonstrate the applicability of qad on ecological datasets.We illustrate that direction and asymmetry in dependence are universal properties of bivariate associations. Qad thus provides additional information gain and avoids model bias and will therefore advance and facilitate the understanding of ecological systems. [ABSTRACT FROM AUTHOR]

This paper aims to provide non‐parametric tests for asymmetric comovements between random variables. We consider the popular Cramér‐von Mises and Kolmogorov–Smirnov test statistics based on the distance between positive and negative joint conditional exceedance distribution functions. These tests can capture both linear and nonlinear dependence in the data and do not require selecting kernel functions and bandwidths. We derive the asymptotic distributions of the tests and establish the validity of a block multiplier‐type bootstrap that one can use in finite‐sample settings. We also show that these tests are consistent for any fixed alternative and have non‐trivial power for detecting local alternatives converging to the null at the parametric rate. Monte Carlo simulations and a real financial data analysis illustrate satisfactory performance of the proposed tests. [ABSTRACT FROM AUTHOR]

This paper examines the unified dissipative performance of Offshore Steel Jacket Platforms (OSJPs) which include actuator failures and probabilistic delays, through a resilient and reliable sampled‐data controller. In detail, random variables fetching the Bernoulli distribution are treated in particular for the stochastic time‐delays. By implementing the Lyapunov–Krasovskii functions(LKF) together with the Second Order Bessel–Legendre Inequality Technique (SOBLIT) and an improved Reciprocally Convex Inequality (RCI) approach, which provides impressive information about sampling to guarantee that the OSJP is unified dissipative. The proposed criterion is exhibited in the sort of linear matrix inequalities (LMIs) and highlights of this study is to examine the multi‐dynamic interpretation of OSJP with extended dissipativity indices. The advantage of the established methodology is explored for OSJPs with the existing results, which is carried out along with the proposed control law, using their values of the experimental parameters. [ABSTRACT FROM AUTHOR]

Recently, many streaming algorithms have utilized generalizations of the fact that the expected maximum distance of any 4‐wise independent random walk on a line over n steps is O(n)$$ O\left(\sqrt{n}\right) $$. In this paper, we show that 4‐wise independence is required for all of these algorithms, by constructing a 3‐wise independent random walk with expected maximum distance Ω(nlgn)$$ \Omega \left(\sqrt{n}\lg n\right) $$ from the origin. We prove that this bound is tight for the first and second moment, and also extract a surprising matrix inequality from these results. Next, we consider a generalization where the steps Xi$$ {X}_i $$ are k‐wise independent random variables with bounded pth moments. We highlight the case k=4,p=2$$ k=4,p=2 $$: here, we prove that the second moment of the furthest distance traveled is O∑Xi2$$ O\left(\sum {X}_i^2\right) $$. This implies an asymptotically stronger statement than Kolmogorov's maximal inequality that requires only 4‐wise independent random variables, and generalizes a recent result of Błasiok. [ABSTRACT FROM AUTHOR]

Simulation is often used in papers and studies across diverse fields like logistics, supply chains, health care, manufacturing, and defense. But simulations must be properly done, including input and model building, designing/analyzing the simulations, and model verification/validation. Unfortunately, simulation studies are not always done well, even though great effort could have gone into the model building and coding. This paper specifies, in brief outline, what authors and researchers need to do, when using simulation as a main tool, to build a convincing case for their findings and conclusions. Considerations on the input side of the model are enumerated, including specification of input distributions and processes; nonstationarity; random-number generation; and generating realizations of random variables and random processes. On the output side are issues of statistical analysis of simulation output; comparison, selection, and ranking of simulated scenarios; variance reduction; and optimum seeking. Involving matters on both the input and output sides are the essential activities of verification and validation. The intent is to establish expectations on what acceptable papers need to do if using simulation, and to serve as a guideline to applied simulation studies. Such papers and studies will then be more valid, more precise, more useful, and ultimately more convincing. [ABSTRACT FROM AUTHOR]

This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR‐type epidemiological models based on discrete‐time Markov chain formulation. The randomization is performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed. [ABSTRACT FROM AUTHOR]

This paper presents a new heuristic for the analysis of adaptive filtering algorithms. It combines Price's Theorem (which is strictly valid when the random variables are jointly Gaussian) with a probabilistic model of the input data that assumes statistical independence between the radial and angular distributions. Moreover, the last distribution is modeled as discrete, which allows obtaining concise and useful closed‐form estimates of the algorithm's performance. The proposed method directly provides a derivation of a traditional result for the steady state performance of the Sign Least Mean Squares algorithm. Furthermore, it can be used to gain new insights into the asymptotic performance of the Block Sign Least Mean Squares algorithm. The analysis results are confirmed through simulations. [ABSTRACT FROM AUTHOR]

The evaluation of a bridge's structural damage state following a seismic event and the decision on whether or not to open it to traffic under the threat of aftershocks (ASs) can significantly benefit from information about the mainshock (MS) earthquake's intensity at the site, the bridge's structural response, and the resulting damage experienced by critical structural components. This paper illustrates a Bayesian network (BN)‐based probabilistic framework for updating the AS risk of bridges, allowing integration of such information to reduce the uncertainty in evaluating the risk of bridge failure. Specifically, a BN is developed for describing the probabilistic relationship among various random variables (e.g., earthquake‐induced ground‐motion intensity, bridge response parameters, seismic damage, etc.) involved in the seismic damage assessment. This configuration allows users to leverage data observations from seismic stations, structural health monitoring (SHM) sensors and visual inspections (VIs). The framework is applied to a hypothetical bridge in Central Italy exposed to earthquake sequences. The uncertainty reduction in the estimate of the AS damage risk is evaluated by utilising various sources of information. It is shown that the information from accelerometers and VIs can significantly impact bridge damage estimates, thus affecting decision‐making under the threat of future ASs. [ABSTRACT FROM AUTHOR]

In this paper, an output‐based event‐triggered control problem of discrete‐time networked control systems (NCSs) subject to bilateral packet dropouts is investigated. In view of stochastic sequences of packet dropouts in the measurement channels (from sensors to controller) and the control channels (from controller to actuators), the NCS is converted into a closed‐loop stochastic parameter system. In the aid of a Lyapunov functional based on stochastic variables, the sufficient conditions on design of a dynamic output feedback control law and bilateral event‐triggering conditions are derived to guarantee the exponentially mean‐square stability for NCSs. Furthermore, an improved iterative algorithm is given to obtain the output‐based control law and event‐triggering parameters from a set of nonconvex inequalities. Finally, a numerical example and the corresponding simulation results are given to illustrate the validity and applicability of the developed techniques. [ABSTRACT FROM AUTHOR]

Most existing methods of multiobjective estimation of distributed algorithms apply the estimation of distribution of the Pareto‐solution on the decision space during the search and little work has proposed on making a regression‐model for representing the final solution set. Some inverse‐model‐based approaches were reported, such as inversed‐model of multiobjective evolutionary algorithm (IM‐MOEA), where an inverse functional mapping from Pareto‐Front to Pareto‐solution is constructed on nondominated solutions based on Gaussian process and random grouping technique. But some of the effective inverse models, during this process, may be removed. This paper proposes an inversed‐model based on random forest framework. The main idea is to apply the process of random forest variable importance that determines some of the best assignment of decision variables (xn) to objective functions (fm) for constructing Gaussian process in inversed‐models that map all nondominated solutions from the objective space to the decision space. In this work, three approaches have been used: classical permutation, Naïve testing approach, and novel permutation variable importance. The proposed algorithm has been tested on the benchmark test suite for evolutionary algorithms [modified Deb K, Thiele L, Laumanns M, Zitzler E (DTLZ) and Walking Fish Group (WFG)] and indicates that the proposed method is a competitive and promising approach. [ABSTRACT FROM AUTHOR]

The time to cover cracking is widely used as a service life indicator in the assessment of deterioration of corrosion‐affected reinforced concrete structures. This paper presents a general probabilistic procedure for prediction of time to cover cracking. Within this procedure, the response surface method (RSM) is employed to calibrate a new model for calculating of radial displacement required for cover cracking based on the results obtained from the finite element (FE) analysis. By taking advantage of the central limit theorem (CLT), simple but accurate probabilistic models for prediction of time to cover cracking that only rely on the knowledge of first and second moments of basic random variables are derived. Rigorous simulation analysis has proved the accuracy of these models. It is shown in the paper that using this probabilistic procedure, factors affecting randomness of time to cover cracking can be easily identified. It is also shown that the time to cover cracking is highly variable with the concrete cover, tensile strength of concrete, corrosion current density, and the model error as the most influential factors on its randomness. [ABSTRACT FROM AUTHOR]