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SIAM J. Sci. Comput. 30, pp. 2207-2234 (28 pages)
Asymptotic Sampling Distribution for Polynomial Chaos Representation from Data: A Maximum Entropy and Fisher Information Approach
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Add to FacebookA procedure is presented for characterizing the asymptotic sampling distribution of estimators of the polynomial chaos (PC) coefficients of a second-order nonstationary and non-Gaussian random process by using a collection of observations. The random process represents a physical quantity of interest, and the observations made over a finite denumerable subset of the indexing set of the random process are considered to form a set of realizations of a random vector $\mathcal{Y}$ representing a finite-dimensional projection of the random process. The Karhunen–Loève decomposition and a scaling transformation are employed to produce a reduced-order model $\mathcal{Z}$ of $\mathcal{Y}$. The PC expansion of $\mathcal{Z}$ is next determined by having recourse to the maximum-entropy principle, the Metropolis–Hastings Markov chain Monte Carlo algorithm, and the Rosenblatt transformation. The resulting PC expansion has random coefficients, where the random characteristics of the PC coefficients can be attributed to the limited data available from the experiment. The estimators of the PC coefficients of $\mathcal{Y}$ obtained from that of $\mathcal{Z}$ are found to be maximum likelihood estimators as well as consistent and asymptotically efficient. Computation of the covariance matrix of the associated asymptotic normal distribution of estimators of the PC coefficients of $\mathcal{Y}$ requires knowledge of the Fisher information matrix (FIM). The FIM is evaluated here by using a numerical integration scheme as well as a sampling technique. The resulting confidence interval on the PC coefficient estimators essentially reflects the effect of incomplete information (due to data limitation) on the characterization of the stochastic process. This asymptotic distribution is significant as its characteristics can be propagated through predictive models for which the stochastic process in question describes uncertainty on some input parameters.
© 2008 Society for Industrial and Applied Mathematics
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History
Received February 14, 2006
Accepted January 16, 2008
Published online June 11, 2008
Accepted January 16, 2008
Published online June 11, 2008
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