Asymptotic Sampling Distribution for Polynomial Chaos Representation of Data: A Maximum Entropy and Fisher information approach

Abstract

A procedure is presented for characterizing the asymptotic sampling distribution of the estimators of the polynomial chaos (PC) coefficients of physical process modeled as non-stationary, non-Gaussian second-order random process by using a collection of observations. These observations made over a denumerable subset of the indexing set of the process are considered to form a set of realizations of a random vector, Y , representing a finite-dimensional model of the random process. The estimators of the PC coefficients of Y are next deduced by relying on its reduced order representation obtained by employing Karhunen-Loève decomposition and subsequent use of the maximum-entropy principle, Metropolis-Hastings Markov chain Monte carlo algorithm and the Rosenblatt transformation. These estimators are found to be maximum likelihood estimators as well as consistent and asymptotically efficient estimators. The computation of the covariance matrix of the associated asymptotic normal distribution of the estimators of these PC coefficients requires evaluation of Fisher information matrix that is evaluated analytically and also estimated by using a sampling technique for the accompanied numerical illustration.Index Terms-Fisher information matrix, Karhunen-Loève expansion, Markov chain Monte Carlo, maximum-entropy probability density estimation, non-Gaussian and nonstationary/non-homogeneous random process/field, polynomial chaos expansion, Rosenblatt transformation.

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