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There is a strong interest in the climate community to improve the ability to accurately predict future trends of climate variables. Recently, machine learning methods have proven their ability to contribute to more accurate predictions of historical data on a variety of climate variables. There is also a strong interest in using statistical downscaling to predict local station data from the output of multi-model ensembles. This project looks at using the machine learning algorithm XGBoost and evaluating its ability to accurately predict historical monthly precipitation, with a focus of applying this method to simulate future precipitation trends. / Thesis / Master of Science (MSc)

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This dissertation focuses on estimation and selection methods for a simple linear model with two levels of variation. This model provides a foundation for extensions to more levels. We propose new regularization criteria for model selection, subset selection, and variable selection in this context. Regularization is a penalized-estimation approach that shrinks the estimate and selects variables for structured data. This dissertation introduces a procedure (HM-ALASSO) that extends regularized multilevel-model estimation and selection to enforce principles of fixed heredity (e.g., including main effects when their interactions are included) and random heredity (e.g., including fixed effects when their random terms are included). The goals in developing this method were to create a procedure that provided reasonable estimates of all parameters, adhered to fixed and random heredity principles, resulted in a parsimonious model, was theoretically justifiable, and was able to be implemented and used in available software. The HM-ALASSO incorporates heredity-constrained selection directly into the estimation process. HM-ALASSO is shown to enjoy the properties of consistency, sparsity, and asymptotic normality. The ability of HM-ALASSO to produce quality estimates of the underlying parameters while adhering to heredity principles is demonstrated using simulated data. The performance of HM-ALASSO is illustrated using a subset of the High School and Beyond (HS&B) data set that includes math-achievement outcomes modeled via student- and school-level predictors. The HM-ALASSO framework is flexible enough that it can be adapted for various rule sets and parameterizations. / Statistics

In this dissertation, a variable selection method for spatial data is developed. It is assumed that the spatial process is non-stationary as a whole but is piece-wise stationary. The pieces where the spatial process is stationary are called regions. The variable selection approach accounts for two sources of correlation: (1) the spatial correlation of the data within the regions, and (2) the correlation of adjacent regions. The variable selection is carried out by including indicator variables that characterize the significance of the regression coefficients. The Ising distribution as prior for the vector of indicator variables, models the dependence of adjacent regions. We present a case study on brook trout data where the response of interest is the presence/absence of the fish at sites in the eastern United States. We find that the method outperforms the case of the probit regression where the spatial field is assumed stationary and isotropic. Additionally, the method outperformed the case where multiple regions are assumed independent of their neighbors. / Ph. D.

As data collection techniques improve, oftentimes the number of covariates exceeds the number of observations. When this happens, regression models become oversaturated and, thus, inestimable. Many classical and Bayesian techniques have been designed to combat this difficulty, with various means of combating the oversaturation. However, these techniques can be tricky to implement well, difficult to interpret, and unstable. What is proposed is a technique that takes advantage of the natural clustering of variables that can often be found in biological and ecological datasets known as the omics datasests. Generally speaking, omics datasets attempt to classify host species structure or function by characterizing a group of biological molecules, such as genes (Genomics), the proteins (Proteomics), and metabolites (Metabolomics). By clustering the covariates and regressing on a single value for each cluster, the model becomes both estimable and stable. In addition, the technique can account for the variability within each cluster, allow for the inclusion of expert judgment, and provide a probability of inclusion for each cluster. / Ph. D.