Variable selection methods with applications
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A data-driven method of generalized adaptive ridge (GAR) for an automatic yet adaptive regression shrinkage and selection was proposed as the first part of this dissertation. In theory, GAR was proved to be equivalent to adaptive- LASSO, ridge and elastic net under appropriate conditions. Simulation results indicated GAR performs either better or equivalent to these methods, in terms of both prediction accuracy and computational speed due to its flexibility of our newly developed algorithm. The second part of this dissertation is on a general adaptive L2-regularized optimization problem for a general loss function. This adaptive L2 penalty term was proved to be equivalent adaptive L1 penalty, adaptive L2 penalty, and combined adaptive L1 and L2 penalty with appropriate choice of parameters and assuming loss function is differentiable. Two algorithms using Newton-Raphson method for the case of the number of predictors (p) less than the number of sample size (n), and sequential minimal optimization (SMO) method for the case p>n and correlated data were developed. The efficacy of this approach was illustrated by simulations, comparisons with other methods and real data analysis. The last part of this dissertation is about adaptive three-way decomposition (ATWD) which combines the adaptive approach in the first part of this dissertation and popular three-way decomposition (TWD) in chemical sciences to analyze nuclear magnetic resonance (NMR) data. This method can be used to reduce the effects of signal noises and dimensionality of the spectral data, provide efficient estimates of spectral components, interpret all the signals retrieved from NMR data, and translate structural information efficiently. Its effective usefulness were illustrated in both simulation studies and real data analysis.