Function Of Several Random Variables Research Articles

Inverse Problem Solution using Bayesian Approach with Application to Darcy Flow Material Parameters Estimation

Standard numerical methods for solving inverse problems in partial differential equations do not reflect a possible inaccuracy in observed data. However, in real engineering applications we cannot avoid uncertainties caused by measurement errors. In the Bayesian approach every unknown or inaccurate value is treated as a random variable. This paper presents an application of the Bayesian inverse approach to the reconstruction of a porosity field as a parameter of the Darcy flow problem. However, this framework can be applied to a wide range of problems that involve some amount of uncertainty. Here the material field is modeled as a Gaussian random field, which is expressed as a function of several random variables. The information about these random variables is given by the resulting posterior distribution, which is then studied using the Cross-Entropy method and samples are generated using the Metropolis-Hastings algorithm.

Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method

The structural reliability analysis is typically based on a model that describes the response, such as maximum deformation or stress, as a function of several random variables. In principle, reliability can be evaluated once the probability distribution of the response becomes available. The paper presents a new method to derive the probability distribution of a function of random variables representing the structural response. The derivation is based on the maximum entropy principle in which constraints are specified in terms of the fractional moments, in place of commonly used integer moments. In order to compute the fractional moments of the response function, a multiplicative form of dimensional reduction method (M-DRM) is presented. Several examples presented in the paper illustrate the numerical accuracy and efficiency of the proposed method in comparison to the Monte Carlo simulation method.

A generalization of Hoeffding’s lemma, and a new class of covariance inequalities

We generalize Hoeffding’s lemma to apply to covariances between functions of several random variables. Our generalization leads to a new class of covariance inequalities involving the Vitali or Hardy–Krause variation. These inequalities are relevant to the study of weakly dependent processes.

A linear approximation model for the parameter design problem

We consider the problem of minimizing the variance of a nonlinear function of several random variables, where the decision variables are the mean values of these random variables. This problem arises in the context of robust design of manufactured products and/or manufacturing processes, where the decision variables are the set points for various design components. A nonlinear programming (NLP) model is developed to obtain approximate solutions for this problem. This model is based on a Taylor series expansion of the given function, up to its linear term. A case study pertaining to the design of a coil spring is discussed, and its corresponding NLP model is developed. Using this model, a non-inferior frontier curve is obtained that shows the trade-off between the minimum mass of the coil spring and the minimum variance of its performance characteristic. Results of applying the model to three other case studies are also presented.

SQVAM: A variance minimizing algorithm

We present an algorithm for minimizing the variance of a nonlinear function of several random variables. Mean values of these random variables are the decision variables, and we allow the variance of each random variable to be a function of its mean value. Potential applications of this algorithm are discussed, and a numerical example is presented.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

Reliability Analysis of Machine Tool Structures

A method of estimating the reliability of machine tool structures is developed. The reliability analysis of horizontal milling machines in various failure modes, like static deflection, fundamental natural frequency and chatter stability, is considered for illustration. The table height, distance of the cutter center from the arbor support, damping factor, Young’s modulus of the material and the load acting on the cutter and the table are considered as random variables. The finite-element displacement method is used to idealize the structure. The reliability analysis is based on the linearization of a function of several random variables about the mean values of the random variables. The overall reliability of the machine tool structure is found by treating it as a weakest-link system having several failure modes. A sensitivity analysis is also conducted to find the variaion of reliability with a change in the coefficients of variation of different random parameters.

Sharp Bounds on the Value of Perfect Information

We present sharp bounds on the value of perfect information for static and dynamic simple recourse stochastic programming problems. The bounds are sharper than the available bounds based on Jensen's inequality. The new bounds use some recent extensions of Jensen's upper bound and the Edmundson-Madansky lower bound on the expectation of a concave function of several random variables. Bounds are obtained for nonlinear return functions and linear and strictly increasing concave utility functions for static and dynamic problems. When the random variables are jointly dependent, the Edmundson-Madansky type bound must be replaced by a less sharp “feasible point” bound. Bounds that use constructs from mean-variance analysis are also presented. With independent random variables the calculation of the bounds generally involves several simple univariate numerical integrations and the solution of several similar nonlinear programs. These bounds may be made as sharp as desired with increasing computational effort. The bounds are illustrated on a well-known problem in the literature and on a portfolio selection problem.

Truncation Errors in First-Order Reliability

In a second moment format for a probabilistic structural design code, it is often necessary to estimate the mean and standard deviation of stress and strength, which in turn may be complicated functions of several random variables. In a broader sense, probabilistic analyses in many disciplines require knowledge of the first and second moments of complicated functions of random variables. This study attempts to describe, under special conditions, the errors introduced in the most widely used approximate method for making this calculation.

A new formalism for the study of configuration-averaged properties of disordered systems

A formalism has been developed for the average of a general function of several random variables. The application of this formalism in problems of configurational averaging in tight-binding models of disordered systems is discussed.

On mixture distributions in pattern recognition

This paper is concerned with the analysis of a class of problems involving mixture variates (functions of several random variables). The explicit inclusion of a mixture as a separate category in pattern recognition problems is studied and plausible forms for the mixing densities in such problems are derived.