In this work. we propose a unified framework called Kalman filter based Radial Basis Functions (KF-RBF) for online functional prediction based on the Radial Basis Functions and the Kalman Filter. The data are nonstationary spatio-ternporal observations irregularly sampled in the spatial domain. We shall assume that a Functional Auto-Regressive (FAR) model is generating the system dynamics. Therefore. to account for the spatial variation. a Radial Basis Function (RBF) network is fitted to the spatial data at every time step. To capture the temporal variation, the regression surfaces arc allowed to change with time. This is achieved by proposing a linear state space model for the RBF weight vectors to evolve temporally. With a fixed functional basis in expressing all regressions. the FAR model call then he re-formulated as a Vector Auto-Regressive (VAR) model embedded in a Kalman Filter. Therefore functional predictions. normally taken place in the Hilbert space. can now be easily implemented 011 a computer. The advantages of our approach are as follows. First it is computationally simple: using the KF. we can obtain the posterior and predictive distributions in closed form. This allows for quick implementation of the model. and provides for full probabilistic inference for the forecasts. Second, the model requires no restrictive assumptions such as stationarity. isotropy or separability of the space/time correlation functions. Third. the method applies to non-lattice data. in which the number and location of sensors can change over time. This framework proposed is further extended by generalizing the real-valued. scalar weights in the functional autoregressive model to operators ill the Reproducing Kernel Hilbert Space (RKHS). This essentially implies that a larger. more intricate class of functions can be represented by this functional autoregressive approach. In other words. the unknown function is expressed as a sum of transformed functions mapped from the past functions in the RKHS. This bigger class of functions can potentially yield a better candidate that is "closer". in the norm sense. to the unknown function. In our research. the KF is used despite the system and observational noise covariance are both unknown. These uncertainties may significantly impact the filter performance. resulting in sub- optimality or divergence. A multiple-model strategy is proposed in view of this. This is motivated by the Interactive Multiple Model (IMM) algorithm in which a collection of filters with different noise characteristics is run in parallel. This strategy avoids the problems associated with the estimation of the noise covariance matrices. Furthermore. it also allows future measurements to be predicted without the assumption of time stationarity of the disturbance terms.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:578701 |
| Date | January 2011 |
| Creators | Su, Jionglong |
| Publisher | University of Sheffield |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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