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Browsing by Subject "High-dimensional data"

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    Unconventional Regression for High-Dimensional Data Analysis
    (2017-06) Gu, Yuwen
    Massive and complex data present new challenges that conventional sparse penalized mean regressions, such as the penalized least squares, cannot fully solve. For example, in high-dimensional data, non-constant variance, or heteroscedasticity, is commonly present but often receives little attention in penalized mean regressions. Heavy-tailedness is also frequently encountered in many high-dimensional scientific data. To resolve these issues, unconventional sparse regressions such as penalized quantile regression and penalized asymmetric least squares are the appropriate tools because they can infer the complete picture of the entire probability distribution. Asymmetric least squares regression has wide applications in statistics, econometrics and finance. It is also an important tool in analyzing heteroscedasticity and is computationally friendlier than quantile regression. The existing work on asymmetric least squares only considers the traditional low dimension and large sample setting. We systematically study the Sparse Asymmetric LEast Squares (SALES) under high dimensionality and fully explore its theoretical and numerical properties. SALES may fail to tell which variables are important for the mean function and which variables are important for the scale/variance function, especially when there are variables that are important for both mean and scale. To that end, we further propose a COupled Sparse Asymmetric LEast Squares (COSALES) regression for calibrated heteroscedasticity analysis. Penalized quantile regression has been shown to enjoy very good theoretical properties in the literature. However, the computational issue of penalized quantile regression has not yet been fully resolved in the literature. We introduce fast alternating direction method of multipliers (ADMM) algorithms for computing penalized quantile regression with the lasso, adaptive lasso, and folded concave penalties. The convergence properties of the proposed algorithms are established and numerical experiments demonstrate their computational efficiency and accuracy. To efficiently estimate coefficients in high-dimensional linear models without prior knowledge of the error distributions, sparse penalized composite quantile regression (CQR) provides protection against significant efficiency decay regardless of the error distribution. We consider both lasso and folded concave penalized CQR and establish their theoretical properties under ultrahigh dimensionality. A unified efficient numerical algorithm based on ADMM is also proposed to solve the penalized CQR. Numerical studies demonstrate the superior performance of penalized CQR over penalized least squares under many error distributions.
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    Variable selection in high-dimensional classification
    (2013-06) Mai, Qing
    Classification has long been an important research topic for statisticians. Nowadays, scientists are further challenged by classification problems for high-dimensional datasets in various fields, ranging from genomics, economics to machine learning. For such massive datasets, classical classification techniques may be inefficient or even infeasible, while new techniques are highly sought-after. My dissertation work tackles high-dimensional classification problems by utilizing variable selection. In particular, three methods are proposed and studied: direct sparse discriminant analysis, semiparametric sparse discriminant analysis and the Kolmogorov filter. In the proposal of direct sparse discriminant analysis (DSDA), I first point out the disadvantage in many current methods that they ignore the correlation structure between predictors. Then DSDA is proposed to extend the well-known linear discriminant analysis to high dimensions, fully respecting the correlation structure. The proposal is efficient and consistent, with excellent numerical performance. In addition to the proposal of DSDA, I also study its connection to many popular proposals of linear discriminant analysis in high dimensions, including the L1-Fisher's discriminant analysis and the sparse optimal scoring. Semiparametric sparse discriminant analysis (SeSDA) extends DSDA by relaxing the normality assumption, which is fundamental for any method requiring the linear discriminant analysis model. SeSDA is more robust than DSDA, while it preserves the good properties of DSDA. Along with the development of SeSDA, a new concentration inequality is obtained that can provide theoretical justifications for methods based on Gaussian copulas. Moreover, the Kolmogorov filter is proposed as a fully nonparametric method that performs variable selection for high-dimensional classification. It requires minimal assumptions on the distribution of the predictors, and is supported by both theoretical and numerical examples. Also, some potential future work is discussed on variable selection in classification.