Learning Methods in Reproducing Kernel Hilbert Space Based on High-dimensional Features

The first topic focuses on the dimension reduction method via the regularization. We propose the selection for principle components via LASSO. This method assumes that some unknown latent variables are related to the response under the highly correlate covariate structure. L 1 regularization plays a key role in adaptively finding a few liner combinations in contrast to the persistent idea that is to employ a few leading principal components. The consistency of regression coefficients and selected model are asymptotically proved and numerical performances are shown to support our suggestion. The proposed method is applied to analyze microarray data and cancer data. Second and third topics focus on the approaches of the independent screening and the dimension reduction with the machine learning approach using positive definite kernels. A Key ingredient matter of these papers is to use reproducing kernel Hilbert space (RKHS) theory. Specifically, we proposed Multiple Projection Model (MPM) and Single Index Latent Factor Model (SILFM) to build an accurate prediction model for clinical outcomes based on a massive number of features. MPM and SILFM can be summarized as three-stage estimation, screening, dimension reduction, and nonlinear fitting. Screening and dimension reduction are unique approaches of two novel methods. The convergence property of the proposed screening method and the risk bound for SILFM are systematically investigated. The results from several simulation scenarios are shown to support it. The proposed method is applied to analyze brain image data and its clinical behavior response.Doctor of Philosoph

Statistical downscaling of air quality models using Principal Fitted Components

Statistical downscaling is a technique that is used to extract high-resolution information from regional scale variables produced by Chemical Transport Models (CTMs). The aim of this thesis is to shade light on the advantages of statistical downscaling in improving the forecasting ability of air quality models. Many statistical downscaling methods in geophysics often rely on dimension reduction techniques to reduce the spatial dimension of gridded model outputs without loss of essential spatial information. In this thesis we developed a new downscaling methodology that relies on using Principal Fitted Components (PFCs) to downscale an air quality model. The main advantage of employing PFCs in downscaling relies in the fact that PFCs represent space-time variations associated with a particular location through the use of inverse regression. This means that PFCs will emphasize on location related regional information. We illustrate our proposed method by both simulation and application on ground level ozone over southeastern U.S region to downscale the Regional ChEmical TrAnsport Model (REAM). Both simulation and applications results indicate that PFC downscaling appears to yield more accurate forecasts. Moreover, we accommodate the fact that covariance matrices that are used to compute PFCs might be unstable due to the fact that they have a relatively large dimension. This issue has motivated us to regularize the covariance matrices by thresholding prior to computing the PFCs and then proceed with the downscaling using thresholded PFCs. We illustrate the modified downscaling approach by simulation and application to ground level ozone. Simulation results suggest that employing thresholded PFCs in downscaling have improved the downscaling results, however, the application results do not agree with the simulation results. Finally, we extend our PFC downscaling method to downscale an ensemble of air quality models. We propose a new two-stage dimension reduction approach to reduce the dimension of an ensemble. The proposed methodology reduces the spatial dimension in each ensemble member, and then the reduced variables are reduced further across the ensemble models. We illustrate our proposed methodology by simulation and application to downscale ground level ozone ensemble outputs in France. Both simulation and application results suggest that our proposed technique seem to show an adequate predictive performance