Large Number Of Random Variables Research Articles

A BIRTH AND DEATH PROCESS FOR BAYESIAN NETWORK STRUCTURE INFERENCE

Bayesian networks are convenient graphical expressions for high-dimensional probability distributions which represent complex relationships between a large number of random variables. They have been employed extensively in areas such as bioinformatics, artificial intelligence, diagnosis, and risk management. The recovery of the structure of a network from data is of prime importance for the purposes of modeling, analysis, and prediction. There has been a great deal of interest in recent years in the NP-hard problem of learning the structure of a Bayesian network from observed data. Typically, one assigns a score to various structures and the search becomes an optimization problem that can be approached with either deterministic or stochastic methods. In this paper, we introduce a new search strategy where one walks through the space of graphs by modeling the appearance and disappearance of edges as a birth and death process. We compare our novel approach with the popular Metropolis–Hastings search strategy and give empirical evidence that the birth and death process has superior mixing properties.

An engineering model for 3-D turbulent wind inflow based on a limited set of random variables

Abstract. Emerging stochastic analysis methods are of potentially great benefit for wind turbine power output and loads analysis. Instead of requiring multiple (e.g. 10 min) deterministic simulations, a stochastic approach can enable a quick assessment of a turbine's long-term performance (e.g. 20-year fatigue and extreme loads) from a single stochastic simulation. However, even though the wind inflow is often described as a stochastic process, the common spectral formulation requires a large number of random variables to be considered. This is a major issue for stochastic methods, which suffer from the curse of dimensionality leading to a steep performance drop with an increasing number of random variables contained in the governing equations. In this paper a novel engineering wind model is developed which reduces the number of random variables by 4–5 orders of magnitude compared to typical models while retaining proper spatial correlation of wind speed sample points across a wind turbine rotor. The new model can then be used as input to direct stochastic simulations models under development. A comparison of the new method to results from the commercial code TurbSim and a custom implementation of the standard spectral model shows that for a 3-D wind field, the most important properties (cross-correlation, covariance, auto- and cross-spectrum) are conserved adequately by the proposed reduced-order method.

Reliability based design optimization with approximate failure probability function in partitioned design space

This paper presents an efficient method for reliability-based design optimization (RBDO), which is robust to complex systems involving computationally expensive numerical models and/or a large number of random variables. This novel method belongs to a type of decoupling approaches in which the failure probability function (FPF) is approximated in the partitioned design space. In the setting of augmented reliability formulation, for a specific design configuration, the failure probability of a system is proportional to the probability density value of design variables conditioned on the failure event, thus transforming FPF approximation into a problem of density estimation. In this paper, we partition the design space into several subspaces and then estimate the density of failure samples in each subspace by binning and constructing regression functions. Sufficient failure samples are efficiently generated in each subspace using Markov Chain Monte Carlo method, which guarantees the accuracy of FPF approximation over there and ultimately over the entire design space. Three illustrative examples involving structural systems subjected to static or dynamic loadings are discussed to demonstrate the efficiency and accuracy of the proposed method.

Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli–Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.

Closed-Form Probabilistic Analysis of Lamination Parameters for Composite Structures

Probabilistic analyses allow predicting the stochastic distribution of an output variable (e.g., the buckling load of a structure) based on the stochastic distribution of input parameters (e.g., material and geometric properties). In the probabilistic analysis of composite structures, one important quantity that can be subject to scatter is the ply or fiber orientation of the layers. Laminates with a large number of plies lead to a large number of random variables, which makes the probabilistic analysis very time consuming. Lamination parameters allow describing any ply layup by a maximum of 12 parameters. They have therefore been used for the design optimization of thick laminates. In the current paper, a two-step procedure is considered for using lamination parameters for probabilistic analyses of composite structures with many plies. In the first step, the stochastic distribution of lamination parameters is determined. In the second step, the actual probabilistic analysis is performed. A closed-form solution for the stochastic moment of lamination parameters is presented. Furthermore, the paper discusses under which circumstances lamination parameters are approximately normally distributed, which allows increasing the efficiency of both steps.

Uncertainty Quantification for an Elasto-acoustic Nonlinear Reduced-Order Computational Model.

The present research concerns the numerical analysis of an uncertain coupled fluid-structure dynamical system, for which the geometrical nonlinearities of the structure induced by the large deformations and the large displacements are taken into account. The structure is coupled with an internal cavity filled with a linear inviscid compressible fluid. The formulation is carried out in terms of displacements and pressure unknowns. The modal basis is made up of the structural modes of the structure and the acoustic modes of the fluid. A nonlinear-reduced order model is then numerically constructed in order to reduce the size of the problem. The uncertainties are then implemented by using the nonparametric probabilistic framework. Note that a peculiar attention that allows the uncertainties on the nonlinear part to be coherently taken into account by preventing the presence of a too large number of random variables used for generating the stochastic model is made. A numerical application is presented.

Modal stability procedure applied to variability in vibration from electromagnetic origin for an electric motor

This paper focuses on the variability in vibration from electromagnetic origin for an electric motor. The multiphysics modeling is introduced by the magnetic force density computation and by the coupling of the 2D electromagnetic and 3D structural dynamic finite element models. The modal stability procedure (MSP) and the Monte Carlo Simulation (MCS) are associated for the variability calculation of natural frequencies and frequency response functions (FRFs) of finite element systems with material random parameters. The MSP formulations are developed for 8-node solid hexahedron element modeling the electric stator. The MCS–MSP approach only requires a single finite element analysis, leading to a fast Monte Carlo simulation using MSP formulation. The validation of MCS-MSP is investigated, the uncertainty propagation is discussed and the computational costs of MCS-MSP are presented. This non-intrusive method provides an evaluation of frequencies and FRFs variability for industrial-size models with a large number of degrees of freedom, a large number of random variables and a low or moderate variability level.

A note on the strong laws of large numbers for random variables

Let \({\{X_n, n \geq1 \}}\) be a sequence of random variables and {bn, n ≥ 1} a nondecreasing sequence of positive constants. No assumptions are imposed on the joint distributions of the random variables. Some sufficient conditions are given under which \({\lim_{n\to \infty}\sum_{i=1}^n X_i/b_n=0}\) almost surely. Necessary conditions for the strong law of large numbers are also given.

On the strong laws of large numbers for weighted sums of random variables

Let {Xn,n≥1} be a sequence of random variables which is stochastically dominated by a random variable. Let {an,n≥1} and {bn,n≥1} be sequences of real numbers with 0<bn↑∞. Sufficient conditions are given under which ∑i=1naiXi/bn→0 almost surely. No conditions are imposed on the joint distributions of the {Xn}. Our results generalize and improve some known results of the strong law of large numbers for random variables. We also give two examples which show the sharpness of our results.

Variational Bayesian strategies for high-dimensional, stochastic design problems

This paper is concerned with a lesser-studied problem in the context of model-based, uncertainty quantification (UQ), that of optimization/design/control under uncertainty. The solution of such problems is hindered not only by the usual difficulties encountered in UQ tasks (e.g. the high computational cost of each forward simulation, the large number of random variables) but also by the need to solve a nonlinear optimization problem involving large numbers of design variables and potentially constraints. We propose a framework that is suitable for a large class of such problems and is based on the idea of recasting them as probabilistic inference tasks. To that end, we propose a Variational Bayesian (VB) formulation and an iterative VB-Expectation-Maximization scheme that is also capable of identifying a low-dimensional set of directions in the design space, along which, the objective exhibits the largest sensitivity. We demonstrate the validity of the proposed approach in the context of two numerical examples involving $\mathcal{O}(10^3)$ random and design variables. In all cases considered the cost of the computations in terms of calls to the forward model was of the order $\mathcal{O}(10^2)$. The accuracy of the approximations provided is assessed by appropriate information-theoretic metrics.

Rare event simulation in finite-infinite dimensional space

Modern engineering systems are becoming increasingly complex. Assessing their risk by simulation is intimately related to the efficient generation of rare failure events. Subset Simulation is an advanced Monte Carlo method for risk assessment and it has been applied in different disciplines. Pivotal to its success is the efficient generation of conditional failure samples, which is generally non-trivial. Conventionally an independent-component Markov Chain Monte Carlo (MCMC) algorithm is used, which is applicable to high dimensional problems (i.e., a large number of random variables) without suffering from ‘curse of dimension’. Experience suggests that the algorithm may perform even better for high dimensional problems. Motivated by this, for any given problem we construct an equivalent problem where each random variable is represented by an arbitrary (hence possibly infinite) number of ‘hidden’ variables. We study analytically the limiting behavior of the algorithm as the number of hidden variables increases indefinitely. This leads to a new algorithm that is more generic and offers greater flexibility and control. It coincides with an algorithm recently suggested by independent researchers, where a joint Gaussian distribution is imposed between the current sample and the candidate. The present work provides theoretical reasoning and insights into the algorithm.

Overlapping Decomposition for Gaussian Graphical Modeling

Correlation based graphical models are developed to detect the dependence relationships among random variables and provide intuitive explanations for these relationships in complex systems. Most of the existing works focus on learning a single correlation based graphical model for all the random variables. However, it is difficult to understand and interpret the massive dependencies of the variables learned from a single graphical model at a global level especially when the graph is large. In order to provide a clearer understanding for the dependence relationships among a large number of random variables, in this paper, we propose the problem of estimating an overlapping decomposition for the Gaussian graphical model of a large scale to generate overlapping sub-graphical models, where strong and meaningful correlations remain in each subgraph with a small scale. Specifically, we propose a greedy algorithm to achieve the overlapping decomposition for the Gaussian graphical model. A key technique of the algorithm is that the problem of solving a $(k+1)$ -node Gaussian graphical model can be approximately reduced to the problem of solving a one-step vector regularization problem based on a solved $k$ -node Gaussian graphical model with theoretical guarantee. Based on this technique, a greedy expansion algorithm is proposed to generate the overlapping subgraphs. Moreover, we extend the proposed method to deal with dynamic graphs where the dependence relationships among random variables vary with the time. We evaluate the proposed methods on synthetic dataset and a real-life traffic dataset, and the experimental results show the superiority of the proposed methods.

Influence of data size on the reliability assessment of creep life of grade 91 steel

Influence of data size on a reliability assessment of the long-term creep life of Grade 91 steel was investigated for two different sets of large body of creep-rupture data comprising 690 (Set-1) and 1072 (Set-2) data points. The Z-parameter on the Larson-Miller parameter of creep rupture data supported a normal distribution well. Based on the normal distribution, the reliability of the predicted creep life for the two data sets was assessed to obtain variations in reliability owing to fluctuations in the service conditions and the dispersion of the creep data using a service condition-creep rupture property interference (SCRI) model. A large number of random variables for Z-parameter were generated using Monte-Carlo simulation. A reliability assessment for the two data sizes under specific service conditions has been comparatively demonstrated. It was found that the size of the creep-rupture data influenced the reliability of the creep life prediction.

Spatial time-dependent reliability analysis of corrosion damage to RC structures with climate change

The environment around concrete structures may be influenced by a changing climate, especially in the long run, leading to an acceleration of deterioration. Therefore, the safety, serviceability and durability of concrete infrastructure may decline at a faster rate than expected. Carbonation-induced deterioration to concrete structures constructed in Sydney, Australia and Kunming, China under a changing climate is investigated in this paper. Two emissions scenarios are considered – RCP 8.5 and RCP 4.5, representing high and medium greenhouse gas emissions scenarios respectively. The spatial time-dependent reliability analysis includes time-dependent climate scenarios and deterioration processes, as well as a large number of random variables and spatial random fields of material properties and dimensions. The surface of concrete structures is discretised into a large number of elements and the likelihood and extent of corrosion damage is calculated by tracking the evolution of the corrosion process of each element using Monte Carlo simulations. The results show that a changing climate could cause the extent of damage to increase by up to 6% for reinforced concrete infrastructure in Kunming. The findings may be used to assess climate adaptation measures in the design stage, as well as a cost–benefit analysis of climate adaptation measures.

A Closed-Form Second-Order Reliability Method Using Noncentral Chi-Squared Distributions

In the second-order reliability method (SORM), the probability of failure is computed for an arbitrary performance function in arbitrarily distributed random variables. This probability is approximated by the probability of failure computed using a general quadratic fit made at the most probable point (MPP). However, an easy-to-use, accurate, and efficient closed-form expression for the probability content of the general quadratic surface in normalized standard variables has not yet been presented. Instead, the most commonly used SORM approaches start with a relatively complicated rotational transformation. Thereafter, the last row and column of the rotationally transformed Hessian are neglected in the computation of the probability. This is equivalent to approximating the probability content of the general quadratic surface by the probability content of a hyperparabola in a rotationally transformed space. The error made by this approximation may introduce unknown inaccuracies. Furthermore, the most commonly used closed-form expressions have one or more of the following drawbacks: They neither do work well for small curvatures at the MPP and/or large number of random variables nor do they work well for negative or strongly uneven curvatures at the MPP. The expressions may even present singularities. The purpose of this work is to present a simple, efficient, and accurate closed-form expression for the probability of failure, which does not neglect any component of the Hessian and does not necessitate the rotational transformation performed in the most common SORM approaches. Furthermore, when applied to industrial examples where quadratic response surfaces of the real performance functions are used, the proposed formulas can be applied directly to compute the probability of failure without locating the MPP, as opposed to the other first-order reliability method (FORM) and the other SORM approaches. The method is based on an asymptotic expansion of the sum of noncentral chi-squared variables taken from the literature. The two most widely used SORM approaches, an empirical SORM formula as well as FORM, are compared to the proposed method with regards to accuracy and computational efficiency. All methods have also been compared when applied to a wide range of hyperparabolic limit-state functions as well as to general quadratic limit-state functions in the rotationally transformed space, in order to quantify the error made by the approximation of the Hessian indicated above. In general, the presented method was the most accurate for almost all studied curvatures and number of random variables.

Fast methods based on modal stability procedure to evaluate natural frequency variability for industrial shell-type structures

This paper proposes a set of parametric numerical methods to predict the effect of uncertainties in the input parameters on the natural frequencies of structures. The first method, called MCS–MSP, involves the Monte Carlo simulation (MCS) and the modal stability procedure (MSP). Here the weak sensitivity of the mode shape to variations in the input parameters of the model is exploited. A single finite element analysis is required for the MCS–MSP, leading to a fast Monte Carlo simulation. Next, two first order methods are presented that rely on the calculation of frequency sensitivity to random variables using either the finite element method (FOFE) or the MSP (FOMSP). These first order methods require only as many finite element or MSP analyses as the number of random variables. These fast and non-intrusive methods are intended to be used with industrial-size models with a large number of degrees of freedom and a large number of random variables. Finally, two applications are presented: a spot welded plate assembly and a car body in white. The stochastic results obtained (mean value, standard deviation, coefficient of variation, statistical distribution) with the three presented methods are compared to those obtained using the direct MCS as a reference. For both examples, the quality of the results obtained with these fast methods is satisfactory. Moreover, the gains are very valuable: the computation time involved in the proposed approaches based on MSP assumption is lower than the computation time needed for six deterministic finite element analyses.

Strong laws of large numbers for weighted sums of asymptotically almost negatively associated random variables

In this article, the strong law of large numbers for weighted sums of asymptotically almost negatively associated (AANA, in short) random variables is obtained. Some sufficient conditions for the strong law of large numbers of random variables are presented. In addition, the results of the paper generalize and improve earlier ones of Chung (Am J Math 69:189–192, 1947) and Teicher (Proc Natl Acad Sci USA 59:705–707, 1968).

Anchored ANOVA Petrov–Galerkin projection schemes for parabolic stochastic partial differential equations

We present a numerical scheme based on the combination of Hoeffding’s functional analysis of variance (ANOVA) decomposition with stochastic Galerkin projection for solving a class of high-dimensional parabolic stochastic partial differential equations (SPDEs). The central idea underpinning the proposed approach is to approximate the SPDE solution using a Hoeffding-ANOVA decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure. We show that when a Dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately, the original stochastic weak formulation can be decoupled into low-dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes. As a result, the proposed approach scales very well to SPDEs with large number of random variables. We theoretically analyze the proposed formulation and provide a priori error estimates, as a function of the spatial, stochastic and temporal discretization parameters and the ANOVA expansion order, that hold under appropriate stochastic regularity assumptions. Numerical studies are presented for a set of time-dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic Galerkin projection scheme (gPC) and the generalized spectral decomposition scheme. These studies show that the proposed approach provides accuracy comparable to classical gPC Galerkin projection schemes while incurring significantly lower computational cost.

Efficient sparse polynomial chaos expansion methodology for the probabilistic analysis of computationally‐expensive deterministic models

SUMMARYThe sparse polynomial chaos expansion (SPCE) methodology is an efficient approach that deals with uncertainties propagation in case of high‐dimensional problems (i.e., when a large number of random variables is involved). This methodology significantly reduces the computational cost with respect to the classical full PCE methodology. Notice however that when dealing with computationally‐expensive deterministic models, the time cost remains important even with the use of the SPCE. In this paper, an efficient combined use of the SPCE methodology and the Global Sensitivity Analysis is proposed to solve such problem. The proposed methodology is firstly validated using a relatively non‐expensive deterministic model that involves the computation of the PDF of the ultimate bearing capacity of a strip footing resting on a weightless spatially varying soil where the soil cohesion and angle of internal friction are modeled by two anisotropic non‐Gaussian cross‐correlated random fields. This methodology is then applied to an expensive model that considers the case of a ponderable soil. A brief parametric study is presented in this case to show the efficiency of the proposed methodology. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Subsampled Gauss Quadrature Nodes for Estimating Polynomial Chaos Expansions

There is a great deal of interest in applying the principles of compressed sensing to problems in uncertainty quantification. One such problem is the approximation of $L^2$ stochastic processes using generalized polynomial chaos expansions. Namely, it presents a new approach to estimating the expansions when the stochastic process is a function of a large number of random variables. This paper describes a new estimation procedure for the Legendre polynomial expansion, within the compressed sensing formalism, by randomly sampling from the set of points defined by Gauss quadrature rules. We give error estimates in the nonasymptotic regime and relate the results to Chebyshev sampling as the number of quadrature points $n\to\infty$. Finally, we compare its real-world performance to other sampling schemes in the literature.