We consider the problem of sparse variable selection on high dimension heterogeneous data sets, which has been taken on renewed interest recently due to the growth of biological and medical data sets with complex, non-i.i.d. structures and prolific response variables. The heterogeneity is likely to confound the association between explanatory variables and responses, resulting in a wealth of false discoveries when Lasso or its variants are na "ively applied. Therefore, the research interest of developing effective confounder correction methods is growing. However, ordinarily employing recent confounder correction methods will result in undesirable performance due to the ignorance of the convoluted interdependency among the prolific response variables. To fully improve current variable selection methods, we introduce a model that can utilize the dependency information from multiple responses to select the active variables from heterogeneous data. Through extensive experiments on synthetic and real data sets, we show that our proposed model outperforms the existing methods.
Recent years have witnessed the great advances in the variable selection area. The most classical approach is @math -norm regularization (i.e. regression @cite_1 ). Further, recent studies have extended the model capability by introducing different regularizers @cite_25 . Examples including the Smoothly Clipped Absolute Deviation (SCAD) @cite_35 , the Local Linear Approximation (LLA) @cite_32 and the Minimax Concave Penalty (MCP) @cite_19 have been introduced recently. These methods overcome the limitations of Lasso @cite_35 at the cost of introducing nonconvexity in the optimization problem. Some other variable selection methods like @cite_7 ignore underlying multidimensional structure, leading to severe small dataset problems. @cite_39 imposes a rank constraint into @math regularization to factor matrice and promotes sparsity in variable selection, which hurts the interpretability. @cite_5 proposed a unsupervised variable selection method, but cannot apply to high dimensional data with heterogeneity.
@cite_35 @cite_7 @cite_1 @cite_32 @cite_39 @cite_19 @cite_5 @cite_25
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Summary. Variable selection plays an important role in high dimensional statistical modelling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality p, accuracy of estimation and computational cost are two top concerns. Recently, Candes and Tao have proposed the Dantzig selector using L1‐regularization and showed that it achieves the ideal risk up to a logarithmic factor log (p). Their innovative procedure and remarkable result are challenged when the dimensionality is ultrahigh as the factor log (p) can be large and their uniform uncertainty principle can fail. Motivated by these concerns, we introduce the concept of sure screening and propose a sure screening method that is based on correlation learning, called sure independence screening, to reduce dimensionality from high to a moderate scale that is below the sample size. In a fairly general asymptotic framework, correlation learning is shown to have the sure screening property for even exponentially growing dimensionality. As a methodological extension, iterative sure independence screening is also proposed to enhance its finite sample performance. With dimension reduced accurately from high to below sample size, variable selection can be improved on both speed and accuracy, and can then be accomplished by a well‐developed method such as smoothly clipped absolute deviation, the Dantzig selector, lasso or adaptive lasso. The connections between these penalized least squares methods are also elucidated. Extracting useful information from high-dimensional data is an important focus of today's statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the L 1 -penalized squared error minimization method Lasso has been popular in regression models and beyond. In this paper, we combine different norms including L 1 to form an intelligent penalty in order to add side information to the fitting of a regression or classification model to obtain reasonable estimates. Specifically, we introduce the Composite Absolute Penalties (CAP) family, which allows given grouping and hierarchical relationships between the predictors to be expressed. CAP penalties are built by defining groups and combining the properties of norm penalties at the across-group and within-group levels. Grouped selection occurs for nonoverlapping groups. Hierarchical variable selection is reached by defining groups with particular overlapping patterns. We propose using the BLASSO and cross-validation to compute CAP estimates in general. For a subfamily of CAP estimates involving only the L 1 and L ∞ norms, we introduce the iCAP algorithm to trace the entire regularization path for the grouped selection problem. Within this subfamily, unbiased estimates of the degrees of freedom (df) are derived so that the regularization parameter is selected without cross-validation. CAP is shown to improve on the predictive performance of the LASSO in a series of simulated experiments, including cases with p » n and possibly mis-specified groupings. When the complexity of a model is properly calculated, iCAP is seen to be parsimonious in the experiments. We propose MC+, a fast, continuous, nearly unbiased and accurate method of penalized variable selection in high-dimensional linear regression. The LASSO is fast and continuous, but biased. The bias of the LASSO may prevent consistent variable selection. Subset selection is unbiased but computationally costly. The MC+ has two elements: a minimax concave penalty (MCP) and a penalized linear unbiased selection (PLUS) algorithm. The MCP provides the convexity of the penalized loss in sparse regions to the greatest extent given certain thresholds for variable selection and unbiasedness. The PLUS computes multiple exact local minimizers of a possibly nonconvex penalized loss function in a certain main branch of the graph of critical points of the penalized loss. Its output is a continuous piecewise linear path encompassing from the origin for infinite penalty to a least squares solution for zero penalty. We prove that at a universal penalty level, the MC+ has high probability of matching the signs of the unknowns, and thus correct selection, without assuming the strong irrepresentable condition required by the LASSO. This selection consistency applies to the case of p » n, and is proved to hold for exactly the MC+ solution among possibly many local minimizers. We prove that the MC+ attains certain minimax convergence rates in probability for the estimation of regression coefficients in l r balls. We use the SURE method to derive degrees of freedom and C p -type risk estimates for general penalized LSE, including the LASSO and MC+ estimators, and prove their unbiasedness. Based on the estimated degrees of freedom, we propose an estimator of the noise level for proper choice of the penalty level. For full rank designs and general sub-quadratic penalties, we provide necessary and sufficient conditions for the continuity of the penalized LSE. Simulation results overwhelmingly support our claim of superior variable selection properties and demonstrate the computational efficiency of the proposed method. We consider the least-square linear regression problem with regularization by the l1-norm, a problem usually referred to as the Lasso. In this paper, we present a detailed asymptotic analysis of model consistency of the Lasso. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection (i.e., variable selection). For a specific rate decay, we show that the Lasso selects all the variables that should enter the model with probability tending to one exponentially fast, while it selects all other variables with strictly positive probability. We show that this property implies that if we run the Lasso for several bootstrapped replications of a given sample, then intersecting the supports of the Lasso bootstrap estimates leads to consistent model selection. This novel variable selection algorithm, referred to as the Bolasso, is compared favorably to other linear regression methods on synthetic data and datasets from the UCI machine learning repository.