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 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. MCMC techniques are applied to one low dimensional chaotic system, the Logistic Map; and to a simplified version of a complex model to represent the UK's electricity grid. In both cases, the technique is implemented in such a way that gradually transform from the PMS case into a more realistic model representation of the system.
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.
More Information can be found at: http://cats.lse.ac.uk/homepages/milena.
Email: milena.cuellar@gmail.com