Huang Jian, Ma Shuangge, Li Hongzhe, Zhang Cun-Hui
Department of Statistics and Actuarial Science, 241 SH University of Iowa Iowa City, Iowa 52242.
Ann Stat. 2011;39(4):2021-2046. doi: 10.1214/11-aos897.
We propose a new penalized method for variable selection and estimation that explicitly incorporates the correlation patterns among predictors. This method is based on a combination of the minimax concave penalty and Laplacian quadratic associated with a graph as the penalty function. We call it the sparse Laplacian shrinkage (SLS) method. The SLS uses the minimax concave penalty for encouraging sparsity and Laplacian quadratic penalty for promoting smoothness among coefficients associated with the correlated predictors. The SLS has a generalized grouping property with respect to the graph represented by the Laplacian quadratic. We show that the SLS possesses an oracle property in the sense that it is selection consistent and equal to the oracle Laplacian shrinkage estimator with high probability. This result holds in sparse, high-dimensional settings with p ≫ n under reasonable conditions. We derive a coordinate descent algorithm for computing the SLS estimates. Simulation studies are conducted to evaluate the performance of the SLS method and a real data example is used to illustrate its application.
我们提出了一种新的用于变量选择和估计的惩罚方法，该方法明确纳入了预测变量之间的相关模式。此方法基于与图相关联的极小极大凹惩罚和拉普拉斯二次型的组合作为惩罚函数。我们将其称为稀疏拉普拉斯收缩（SLS）方法。SLS使用极小极大凹惩罚来鼓励稀疏性，并使用拉普拉斯二次惩罚来促进与相关预测变量相关的系数之间的平滑性。SLS相对于由拉普拉斯二次型表示的图具有广义分组属性。我们表明，SLS具有一种神谕属性，即它在选择上是一致的，并且在高概率下等于神谕拉普拉斯收缩估计量。在合理条件下，该结果在(p\gg n)的稀疏高维设置中成立。我们推导了一种用于计算SLS估计量的坐标下降算法。进行了模拟研究以评估SLS方法的性能，并使用一个实际数据示例来说明其应用。
