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Perspectives and advances in parameter estimation of nonlinear models.

Cuellar Sanchez, Milena Clarissa (2007) Perspectives and advances in parameter estimation of nonlinear models. PhD thesis, London School of Economics and Political Science (United Kingdom).

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Abstract

Nonlinear methodologies to estimate parameters of deterministic nonlinear models are investigated in the case where experimental observations are available and uncertainty sources are present, e.g. model inadequacy, model error and noise. The problem of parameter estimation is interpreted from a nonlinear dynamical time series analysis perspective; however deterministic and probabilistic techniques originated outside the nonlinear deterministic framework are studied, implemented and discussed. Conceptually, the Thesis is divided in two parts that explore two fundamentally different approaches: (a) Bayesian and (b) Geometrical estimation. Both approaches attempt to estimate parameters and model states in the case where the system and the model used to represent it are identical, i.e. Perfect Model Scenario (PMS), even though the implications of the results obtained are considered for Imperfect Model cases. The performance of the resulting model parameter estimates in control monitoring and forecasting of the corresponding system is assessed in an application-oriented fashion and contrasted where possible with system observations, in order to look for a consistent way to combine probabilistic and deterministic approaches. Given the presence of uncertainty in the model used to represent a system and in the observations available, combined methodologies enable us to best interpret the resulting estimates in a probabilistic framework as well as in the context of a particular application. The first conceptual part relates to the REMIND project, which is to find a way to meld advances in nonlinear dynamics with those in Bayesian estimation for both mathematical systems and real industrial settings, i.e. for control monitoring the UK's electricity grid system efficiently. Bayesian inference is used to estimate model parameters and model states using Markov Chain Monte Carlo (MCMC) techniques. For the observations of grid frequency and demand, the operational constraints of the data sets are maintained through the estimation process, for example in the situation where the data are provided at rates that restrict on-line storage and post processing. When MCMC is applied to the Logistic Map, curious behaviour of the convergence of the Markov Chain and in the resulting parameter and states estimates are observed and are suspected to be a consequence of high multimodality in the resulting posterior, which in turn generates estimates with a low dynamical informational content. In the case when the MCMC is applied to a UK's grid frequency dynamical model, the technique is implemented in such a way that gradually transform from the PMS case into a more realistic model representation of the system. Convergence of the MCMC algorithm for the grid frequency models is highly dependent on the quality of operational data, which fails to provide the information required by the tailor-made MCMC implementation. In addition, sanity checks are proposed to establish meaningful convergence of MCMC analyses of time series in general. The second conceptual part explores a new approach to parameter estimation in nonlinear modelling, based on the geometric properties of short term model trajectories, whilst keeping track of the global behaviour of the model. Geometric properties are defined in the context of indistinguishable states theory. Parameter estimates are found for low dimensional chaotic systems by means of Gradient Descent methods (GD) in the PMS. Some of the advances are made possible by means of improving the balance between information extracted from the observations and from the dynamical equations. As a result of this investigation, it is noted that, even with perfect knowledge of system and noise in both models, the uncertainty in the dynamics cannot be distinguished from the uncertainty in the observations. In addition, the Geometric approach and Bayesian approach of the problem of model parameter and state estimation for nonlinear models in the PMS are compared aiming to distinguish them based on dynamical features of the estimates. In the Bayesian formulation there are still fundamental challenges when a perfect model is not available.

Item Type: Thesis (PhD)
Uncontrolled Keywords: Statistics
Sets: Collections > ProQuest Etheses
Departments > Statistics
URI: http://etheses.lse.ac.uk/id/eprint/2712

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