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A new statistical model designed for regression analysis with a sparse design matrix is proposed. This new model utilizes the positions of the limited non-zero elements in the design matrix to decompose the regression model into sub-regression models. Statistical inferences are further made on the values of these limited non-zero elements to provide a reference for synthesizing these sub-regression models. With this concept of the regression decomposition and synthesis, the information on the structure of the design matrix can be incorporated into the regression analysis to provide a more reliable estimation. The proposed model is then applied to resolve the spatial resolution enhancement problem for spatially oversampled images. To systematically evaluate the performance of the proposed model in enhancing the spatial resolution, the proposed approach is applied to the oversampled images that are reproduced via random field simulations. These application results based on different generated scenarios then conclude the effectiveness and the feasibility of the proposed approach in enhancing the spatial resolution of spatially oversampled images.

By recognizing that the main difficulty of the modeling of daily precipitation amounts is the selection of an appropriate probability distribution, this study aims to establish a model selection framework to identify the appropriate probability distribution for the modeling of daily precipitation amounts from the commonly used probability distributions, i.e. the exponential, Gamma, Weibull, and mixed exponential distributions. The mixed Gamma Weibull (MGW) distribution serves this purpose because all the commonly used probability distributions are special cases of the MGW distribution, and the MGW distribution integrates all the commonly used probability distributions into one framework. Finally, via the large sample inference of likelihood ratios, a model selection criterion can be established to identify the appropriate model for the modeling of daily precipitation amounts from the MGW distribution framework.

A new stochastic model for daily precipitation occurrence processes observed at multiple locations is developed. The modeling concept is to use the indicator function and the elliptical shape of multivariate Gaussian distribution to represent the joint probabilities of daily precipitation occurrences. By using this concept, the number of parameters needed for precipitation occurrence modeling can be largely reduced when compared to the commonly used two-state Markov chain approach. With this parameter reduction, the modeling of spatio-temporal dependence of daily precipitation occurrence processes observed at different locations is no longer difficult. Results of an illustrative application using the precipitation record available from a network of ten raingauges in the southern Quebec region, also demonstrate the accuracy and the feasibility of the proposed model.