We present a novel approach to parameter estimation of systems with complicated dynamics, as well as evidence for the existence of a universal power law that enables us to quantify the dependence of global geometry on small changes in the parameters of the system. This power law gives rise to what seems to be a new dynamical system invariant.

Mathematical models often involve unknown parameters that must be fit to experimental data. These so-called parameter estimation problems have many applications that may involve differential equations, optimization, and control theory. EASY-FIT and SOCS are two software packages that solve parameter estimation problems. In this thesis, we discuss the design and implementation of a source-to-source translator called EFtoSOCS used to translate EASY FIT input into SOCS input. This makes it possible to test SOCS on a large number of parameter estimation problems available in the EASY-FIT problem database that vary both in size and difficulty.<p>Parameter estimation problems typically have many locally optimal solutions, and the solution obtained often depends critically on the initial guess for the solution. A 3-stage approach is followed to enhance the convergence of solutions in SOCS. The stages are designed to use an initial guess that is progressively closer to the optimal solution found by EASY-FIT. Using this approach we run EFtoSOCS on all translatable problems (691) from the EASY-FIT database. We find that all but 7 problems produce converged solutions in SOCS. We describe the reasons that SOCS was not able solve these problems, compare the solutions found by SOCS and EASY-FIT, and suggest possible improvements to both EFtoSOCS and SOCS.

Mathematical models often involve unknown parameters that must be fit to experimental data. These so-called parameter estimation problems have many applications that may involve differential equations, optimization, and control theory. EASY-FIT and SOCS are two software packages that solve parameter estimation problems. In this thesis, we discuss the design and implementation of a source-to-source translator called EFtoSOCS used to translate EASY FIT input into SOCS input. This makes it possible to test SOCS on a large number of parameter estimation problems available in the EASY-FIT problem database that vary both in size and difficulty.<p>Parameter estimation problems typically have many locally optimal solutions, and the solution obtained often depends critically on the initial guess for the solution. A 3-stage approach is followed to enhance the convergence of solutions in SOCS. The stages are designed to use an initial guess that is progressively closer to the optimal solution found by EASY-FIT. Using this approach we run EFtoSOCS on all translatable problems (691) from the EASY-FIT database. We find that all but 7 problems produce converged solutions in SOCS. We describe the reasons that SOCS was not able solve these problems, compare the solutions found by SOCS and EASY-FIT, and suggest possible improvements to both EFtoSOCS and SOCS.

Thesis (Ph. D.)--Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 2001. / "Funding was provided by the Office of Naval Research under grant N00014-01-1-0257 and the WHOI Education Office." "June, 2001." Includes bibliographical references (p. 207-215).

Thesis (M.S. in Aeronautical Engineering)--Naval Postgraduate School, September 1995. / Thesis advisor(s): Daniel J. Collins, Rocjard M. Howard. Includes bibliographical references (p. 83-84).

Motivated by problems in medicine, biology, engineering and economics, con- strained parameter problems arise in a wide variety of applications. Among them the application to the dose-response of a certain drug in development has attracted much interest. To investigate such a relationship, we often need to conduct a dose- response experiment with multiple groups associated with multiple dose levels of the drug. The dose-response relationship can be modeled by a shape-restricted normal regression. We develop an iterative two-step ascent algorithm to estimate normal means and variances subject to simultaneous constraints. Each iteration consists of two parts: an expectation{maximization (EM) algorithm that is utilized in Step 1 to compute the maximum likelihood estimates (MLEs) of the restricted means when variances are given, and a newly developed restricted De Pierro algorithm that is used in Step 2 to find the MLEs of the restricted variances when means are given. These constraints include the simple order, tree order, umbrella order, and so on. A bootstrap approach is provided to calculate standard errors of the restricted MLEs. Applications to the analysis of two real datasets on radioim-munological assay of cortisol and bioassay of peptides are presented to illustrate the proposed methods. Liu (2000) discussed the maximum likelihood estimation and Bayesian estimation in a multinomial model with simplex constraints by formulating this constrained parameter problem into an unconstrained parameter problem in the framework of missing data. To utilize the EM and data augmentation (DA) algorithms, he introduced latent variables {Zil;Yil} (to be defined later). However, the proposed DA algorithm in his paper did not provide the necessary individual conditional distributions of Yil given (the observed data and) the updated parameter estimates. Indeed, the EM algorithm developed in his paper is based on the assumption that{ Yil} are fixed given values. Fortunately, the EM algorithm is invariant under any choice of the value of Yil, so the final result is always correct. We have derived the aforesaid conditional distributions and hence provide a valid DA algorithm. A real data set is used for illustration. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Parameter estimation plays a key role in many signal processing applications. Traditional parameter estimation relies on centralized method which requires gathering of all information dispersed over the network in a central processing unit. As the scale of network increases, centralized estimation is not preferred since it requires not only the knowledge of network topology but also heavy communications from peripheral nodes to central processing unit. Besides, computation at the control center cannot scale indefinitely with the network size. Therefore, distributed estimation which involves only local computation at each node and limited information exchanges between immediate neighbouring nodes is needed. In this thesis, for local observations in the form of a pairwise linear model corrupted by Gaussian noise, belief propagation (BP) algorithm is investigated to perform distributed estimation. It involves only iterative updating of the estimates with local message exchange between immediate neighboring nodes. Since convergence has always been the biggest concern when using BP, we establish the convergence properties of asynchronous vector form Gaussian BP under the pairwise model. It is shown analytically that under mild condition, the asynchronous BP algorithm converges to the optimal estimates with estimation mean square error (MSE) at each node approaching the centralized Bayesian Cram´er-Rao bound (BCRB) regardless of the network topology. The proposed framework encompasses both classes of synchronous and asynchronous algorithms for distributed estimation and is robust to random link failures. Two challenging parameter estimation problems in large-scale networks, i.e., network-wide distributed carrier frequency offsets (CFOs) estimation, and global clock synchronization in sensor network, are studied based on BP. The proposed algorithms do not require any centralized information processing nor knowledge of the global network topology and are scalable with the network size. Simulation results further verify the established theoretical analyses: the proposed algorithms always converge to the optimal estimates regardless of network topology. Simulations also demonstrate the MSE at each node approaches the corresponding centralized CRB within a few iterations of message exchange. Furthermore, distributed estimation is studied for the linear model with unknown coefficients. Such problem itself is challenging even for centralized estimation as the nonlinear property of the observation model. One problem following this model is the power state estimation with unknown sampling phase error. In this thesis, distributed estimation scheme is proposed based on variational inference with parallel update schedule and limited message exchange between neighboring areas, and the convergence is guaranteed. Simulation results show that after convergence the proposed algorithm performs very close to that of the ideal case which assumes perfect synchronization, and centralized information processing. / published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

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Thesis (PhD)--University of South Australia, 2000

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.