This thesis concerns the solutions to the ill-posed problem in functional regression, where either covariates or responses are in functional spaces. The regression coefficient in these functional settings lives on infinite-dimensional spaces. Therefore, dimension reduction is commonly considered. In Chapter 2, to analyze trends in stratospheric ozone profiles, the profiles are regressed on the time and relevant proxies (function on multivariate regression). To achieve dimension reduction, we employ Functional Principal Component Analysis (FPCA) and the projections of the profiles onto the PC basis are used in the subsequent statistical step to reveal the non-linear effects of the covariates on ozone. Variations in the influences and the trends across altitudes are found, which highlights the benefits of using the functional approach. When the PC basis is used for the regression coefficient, the subspace is chosen without regard to how well it helps prediction. In Chapter 3, we introduce a more efficient dimension reduction method, Functional Principal Fitted Component Regression (FPFCR), accounting for the response when choosing the components, based on an inverse regression. Our numerical studies provide insights about the possible advantages of using our proposed approach: it leads to more parsimonious model selection, compared to classical dimension reduction methods, which is particularly apparent in our brain image analysis. The solutions to the regression problem above are based on a frequentist perspective. In Chapter 4, we adopt a Bayesian viewpoint and propose Functional Bayesian Linear Regression (FBLR). We impose a Gaussian prior for the regression coefficient with the precision written in differential form. In addition, a Gamma prior is assumed for the precision of the regression error. We obtain the posterior of the regression parameter and further quantify its uncertainties via point-wise Bayesian credible regions.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:634642 |
| Date | January 2014 |
| Creators | Park, A. Y. |
| Publisher | University College London (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://discovery.ucl.ac.uk/1455629/ |
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