Parameter Selection Methods in Inverse Problem Formulation
We discuss methods for a priori selection of parameters to be estimated in inverse problem formulations (such as Maximum Likelihood, Ordinary and Generalized Least Squares) for dynamical systems with numerous state variables and an even larger number of p
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Parameter Selection Methods in Inverse Problem Formulation H.T. Banks, Ariel Cintr´on-Arias, and Franz Kappel
Abstract We discuss methods for a priori selection of parameters to be estimated in inverse problem formulations (such as Maximum Likelihood, Ordinary and Generalized Least Squares) for dynamical systems with numerous state variables and an even larger number of parameters. We illustrate the ideas with an in-host model for HIV dynamics which has been successfully validated with clinical data and used for prediction and a model for the reaction of the cardiovascular system to an ergometric workload.
H.T. Banks () Center for Research in Scientific Computation and Center for Quantitative Sciences in Biomedicine, Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8212 e-mail: [email protected] A. Cintr´on-Arias Center for Research in Scientific Computation and Center for Quantitative Sciences in Biomedicine, Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8212 Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-0663 e-mail: [email protected] F. Kappel Center for Research in Scientific Computation and Center for Quantitative Sciences in Biomedicine, Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8212 Institute for Mathematics and Scientific Computation, University of Graz, A8010 Graz, Austria e-mail: [email protected] J.J. Batzel et al. (eds.), Mathematical Modeling and Validation in Physiology, Lecture Notes in Mathematics 2064, DOI 10.1007/978-3-642-32882-4 3, © Springer-Verlag Berlin Heidelberg 2013
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3.1 Introduction There are many topics of great importance and interest in the areas of modeling and inverse problems which are properly viewed as essential in the use of mathematics and statistics in scientific inquiries. A brief, noninclusive list of topics include the use of traditional sensitivity functions (TSF) and generalized sensitivity functions (GSF) in experimental design (what type and how much data is needed, where/when to take observations) [9–11, 16, 56], choice of mathematical models and their parameterizations (verification, validation, model selection and model comparison techniques) [8, 12, 13, 17, 21–24, 41], choice of statistical models (observation process and sampling errors, residual plots for statistical model verification, use of asymptotic theory and bootstrapping for computation of standard errors, confidence intervals) [8, 14, 30, 31, 54, 55], choice of cost functionals (maximum likelihood estimation, ordinary least squares, weighted least squares, generalized least squares, etc.) [8, 30], as well as parameter identifiability and selectivity. There is extensive literature on each of these topics and many have been treated in surveys in one form or another ([30] is an excellent monograph with many references on the statistically related topics) or in earlier lecture notes [8]. We discuss here an enduring majo
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