Uncertainty Quantification in the Estimation of Probability Distributions on Parameters in Size-Structured Population Models

We consider ordinary least squares (OLS) parameter estimation problems in which the underlying dynamics are described by partial differential equations and the unknown parameter of interest is a probability distribution describing the variability of growth rates across a size-structured population. The focus of our work is the development of an inverse problem computational methodology for the estimation of functional parameters in the presence of (model and data) uncertainty. Since this optimization problem involves both an infinite-dimensional state space and an infinite-dimensional parameter space, computationally efficient approximation methods for both parametric and non-parametric versions of the OLS inverse problem are developed and discussed. The approximation methods that we present are applicable to a variety of inverse problems, including Type II problems in which only aggregate or population level longitudinal data is available. We compare computational and statistical results of a delta function approximation method, a spline based approximation method, and a standard parametric OLS formulation. The latter uses an a priori probability distribution in the inverse problem for estimation of distributions of growth rates in size-structured marine populations. After summarizing the underlying theoretical framework, we present several numerical examples as validation of the theory. Convergence as well as sensitivity of the estimates with respect to noise in the data is discussed for both approximation methods. A computational framework for quantification of uncertainty associated with the estimated parameters is given and sample numerical findings are presented. We demonstrate how to construct "functional" confidence bands that will aid in quantifying the uncertainty in estimated probability distributions by extending the standard asymptotic theory for finite-dimensional OLS estimators. Using our inverse problem methodology, we present results for the estimation of growth rate distributions in size-structured marine populations illustrating the strengths and weaknesses associated with the three different computational schemes.

Author dc:creator

• Davis, Jimena Lamanda

Advisors dc:contributor.advisor

• Dr. Marie Davidian, Committee Member
• Dr. Alun L. Lloyd, Committee Member
• Dr. H.T. Banks, Committee Chair
• Dr. Hien T. Tran, Committee Member

• Inverse Problem
• Probability Distributions
• Confidence Bands
• Uncertainty
• Size-Structured Population Models

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Dc Identifier Other

etd-07182008-151056

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https://repository.lib.ncsu.edu/server/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Arepository.lib.ncsu.edu%3A1840.16%2F5401