Fused Lasso and Tensor Covariance Learning with Robust Estimation

With the increase in computation and data storage, there has been a vast collection of information gained with scientific measurement devices. However, with this increase in data and variety of domain applications, statistical methodology must be tailored to specific problems. This dissertation is focused on analyzing chemical information with an underlying structure. Robust fused lasso leverages information about the neighboring regression coefficient structure to create blocks of coefficients. Robust modifications are made to the mean to account for gross outliers in the data. This method is applied to near infrared spectral measurements in prediction of an aqueous analyte concentration and is shown to improve prediction accuracy. Expansion on the robust estimation and structure analysis is performed by examining graph structures within a clustered tensor. The tensor is subjected to wavelet smoothing and robust sparse precision matrix estimation for a detailed look into the covariance structure. This methodology is applied to catalytic kinetics data where the graph structure estimates the elementary steps within the reaction mechanism.