Two advancements in statistical learning methods

This thesis contributes significantly to data science by advancing methods for analyzing high-dimensional datasets, particularly focusing on key relationships between outcomes and predictor variables. It introduces two novel statistical learning methods: missoNet and DKLasso (including DKLasso+). These methods address two main challenges: managing incomplete data and enhancing the interpretability of deep learning models.First, missoNet is introduced as a sparse multi-task regression method designed specifically for high-dimensional datasets with missing values. Traditional multi-task regression often struggles with incomplete data, leading to biased or inefficient results. missoNet overcomes this by treating observed data as noisy representations of true values and using a modified optimization objective that remains convex even with missing entries. It also applies L1 regularization to ensure stable estimates, allowing missoNet to provide robust and accurate results, even in complex high-dimensional settings.The effectiveness of missoNet is demonstrated using two datasets. The first analyzes anti-citrullinated protein antibody (ACPA) related DNA methylation and genetic data to identify methylation quantitative trait loci (mQTLs), focusing on methylation and SNPs within the same genomic region. The second example uses data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) to estimate associations between FDG-PET measurements across 90 brain regions and factors such as cerebrospinal fluid biomarkers. These applications show how missoNet effectively handles incomplete high-dimensional data, offering valuable insights for complex biological studies.The thesis then introduces DKLasso and DKLasso+, which extend the deep kernel learning (DKL) framework to improve model interpretability through feature sparsity. While deep learning excels at identifying intricate patterns, it often lacks transparency. DKLasso tackles this by adding a linear residual connection with an L1 penalty to the DKL model, promoting explicit feature selection by deactivating irrelevant input variables. This approach improves interpretability while maintaining the nonparametric flexibility of kernel methods, which also helps to provide reliable uncertainty estimates - a critical feature in scientific research where confidence in predictions is paramount.The efficacy of DKLasso and DKLasso+ is illustrated using data from the Canadian Longitudinal Study on Aging (CLSA), where these methods identified key metabolites influencing the acceleration of epigenetic aging. This example highlights the models' strength in uncovering meaningful patterns in datasets with complex interactions.Together, missoNet and DKLasso (including DKLasso+) address critical challenges in linear and nonlinear modeling, providing powerful tools for high-dimensional data analysis. These methods are not only robust and capable, but also designed to be accessible for researchers and practitioners aiming to apply advanced data analysis techniques

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