Gaussian Process Regression for Model Estimation

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State estimation techniques using Kalman filter and Particle filters are used in a number of applications like tracking, econometrics, weather data assimilation, etc. These techniques aim at estimating the state of the system using the system characteristics. System characteristics include the definition of system's dynamical model and the observation model. While the Kalman filter uses these models explicitly, particle filter based estimation techniques use these models as part of sampling and assigning weights to the particles. If the state transition and observation models are not available, an approximate model is used based on the knowledge of the system. However, if the system is a total black box, it is possible that the approximate models are not the correct representation of the system and hence will lead to poor estimation.

This thesis proposes a method to deal with such situations by estimating the models and the states simultaneously. The thesis concentrates on estimating the system's dynamical model and the states, given the observation model and the noisy observations. A Gaussian process regression based method is developed for estimating the model. The regression method is sped up from O(N2) to O(N) using an data-dependent online approach for fast Gaussian summations. A relevance vector machine based data selection scheme is used to propagate the model over iterations. The proposed method is tested on a Local Ensemble Kalman Filter based estimation for the highly non-linear Lorenz-96 model.