Synthesis of Discrete Distributed, Correlated Multivariates Utilizing Walsh Functions for Uncertainty Quantification

摘要

A variate, which may be represented by a probability mass function, is intended to be equivalent to a probability density function of a continuous random variable. The variates are composed of a discrete set of values and associated weights. The sum of the individual elements of the array, each multiplied by its associated weight, represents the expected value, or Expectation, of the array. Further, the accuracy of the variate may be determined by comparison of statistical to analytical nth order moments, or moments of specific functions of the variable for which analytic solutions are known. For quadrature, such as Gauss-Hermite for normal distribution and Gauss-Legendre for uniform distribution, a specified number of weights and values are determined from roots of the associated polynomial resulting in highly accurate approximations. Alternatively, a set of values with equal weights are typically produced through random number generation and converge to the expected value as their population increases. A method is presented to decompose a population of a variate by using a linear function of two-point distributions, based on Walsh functions, resulting in a unique set of coefficients. A specific distribution may be formulated by optimizing the coefficients of the linear function to attain the moments, raw or central, that correspond with the desired stochastic characteristics. Multivariate populations that are independent may be assembled from individual synthesized distributions, and correlated multivariate populations may be formulated through simultaneous optimization of multiple distributions using additional constraints applied to the covariances of the products of variates. A known function of specified distribution is examined to compare accuracy and efficiency of statistical analysis performed on formulated populations to random populations using the Monte Carlo method. Uncertainty Quantification of a canonical structure problem using Finite Element Analysis illustrates the ability of formulated populations to capture stochastic characteristics in comparison to larger random populations.