Sparse group sufficient dimension reduction and covariance cumulative slicing estimation
"This dissertation contains two main parts: In Part One, for regression problems with grouped covariates, we adopt the idea of sparse group lasso (Friedman et al., 2010) to the framework of the sufficient dimension reduction. We propose a method called the sparse group sufficient dimension reduction (sgSDR) to conduct group and within group variable selections simultaneously without assuming a specific model structure on the regression function. Simulation studies show that our method is comparable to the sparse group lasso under the regular linear model setting, and outperforms sparse group lasso with higher true positive rates and substantially lower false positive rates when the regression function is nonlinear or (and) the error distributions are non-Gaussian. One immediate application of our method is to the gene pathway data analysis where genes naturally fall into groups (pathways). An analysis of a glioblastoma microarray data is included for illustration of our method. In Part Two, for many-valued or continuous Y , the standard practice of replacing the response Y by a discrete version of Y usually results in the loss of power due to the ignorance of intra-slice information. Most of the existing slicing methods highly reply on the selection of the number of slices h. Zhu et al. (2010) proposed a method called the cumulative slicing estimation (CUME) which avoids the otherwise subjective selection of h. In this dissertation, we revisit CUME from a different perspective to gain more insights, and then refine its performance by incorporating the intra-slice covariances. The resulting new method, which we call the covariance cumulative slicing estimation (COCUM), is comparable to CUME when the predictors are normally distributed, and outperforms CUME when the predictors are non-Gaussian, especially in the existence of outliers. The asymptotic results of COCUM are also well proved."--Abstract, page iv.
Missouri University of Science and Technology
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