充分降维理论和方法的拓展研究
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摘要
这篇论文致力于对充分降维领域中一些理论的深入研究以及方法上的延伸。
在充分降维领域中有两个重要的话题。第一个是估计中心(均值)降维空间的基方向。而经典降维方法基方向样本估计的大样本性质至今仍不明了。为了进一步了解这些常用降维方法特别是其方向估计的理论性质,在这篇论文中我们首先研究了这些方法的核矩阵以及基方向样本估计的二阶渐近性质。我们推导了四种常见的降维方法,包括切片逆回归(Sliced Inverse Regression, SIR, Li,1991),切片平均方差估计(Sliced Average Variance Estimation, Cook and Weisberg,1991),海赛主方向(Principal Hessian Direction, Li,1992)和方向回归(Directional Regression, Li and Wang,2007),它们样本估计的二阶渐近展开式。利用这些降维方法的二阶渐近展式,我们可以进一步考虑纠正其O(n-1)偏差以提高估计的精度。从已经得到的二阶渐近展式中,我们可以求出降维方法方向估计二阶偏差的显示表达式,继而可以很容易得到二阶偏差的相合样本估计。我们随后提出一种一般的降维方法的偏差纠正策略,其思想很简单：即是将某一种降维方法基方向的样本估计减去其二阶偏差的样本估计。并且我们证明了经过偏差纠正后,这些降维方法方向估计的偏差被缩小到O(n-2)。
充分降维领域中的另一个重要的问题是决定中心(均值)降维子空间的结构维数。常用的选取结构维数的方法都有其局限性。序贯检验法依赖于检验的显著性水平。重抽样方法的运算量过大。Zhu, Miao and Peng (2006)提出的BIC准则虽然可以相合的估计结构维数,但如何基于数据选取最优的惩罚函数是一个难题。更重要的是,一般充分降维的过程分为两步,首先是决定维数,然后再选取相应的基方向。论文的第二部分基于对降维方法核矩阵样本特征值的压缩估计提出一种稀疏谱分解方法用以决定结构维数。该方法的主要思路是通过建立矩阵谱分解与最小二乘之间的联系,然后利用Zou(2006)所提出的自适应性最小绝对缩减和变量选择算子得到样本特征值的稀疏估计。和以往降维方法的两步估计不同,稀疏谱分解方法可以同时估计结构维数和中心(均值)降维子空间的基方向。同时我们还证明了稀疏谱分解方法具有Oracle性质。
本文的第三部分是将非参数方法B样条用以估计降维方法SIR和SAVE的核矩阵。和已有的切片方法以及核估计方法相比,B样条方法估计精度更高并且也同样计算简单。另外我们修正了Zhu, Miao and Peng (2006)所提出的BIC准则。修正的目的是为了平衡BIC准则中的主项与惩罚项,使得其数量级大致相仿。这种修正的BIC准则在估计结构维数方面的精度较之传统方法也有所提高。
对于半参数模型的降维一般有两个出发点：一是找出模型中可能存在的变量的线性组合,二是选择模型中的重要变量。其中第一点就是充分降维的概念。而第二点是当前统计学界非常热门的话题：变量选择。本文的第四部分提出一种同时进行充分降维和变量选择的新方法。受到Candes and Tao(2007)一文的启发,我们借鉴DantzigSelector对SIR方法进行了e1规范化。这种新方法的本质是在从SIR的谱分解形式中求解基方向与极小化基方向e1模长之间的一种折中方案。我们所提出的这种新的充分降维与变量选择相结合的方法,在样本量n小于变量维数p的情形下仍然行之有效。当变量维数p固定时,我们得到了这种新方法的相合性和渐近正态性。当变量维数p和n以一定形式趋向于无穷时(需要指出的是p>n的情形只是这种形式的特殊情形),我们得到了这种方法估计的误差上界。
最后我们基于方向回归提出一种不基于模型假设的变量显著性检验。我们推导了在原假设下所提出的统计量的渐近分布。另外基于统计量的渐近性质,我们提出了两种非常简单的不基于模型假设的变量选择方法。这两种新方法不同于当前流行的变量选择方法比如最小绝对缩减和变量选择算子(Least Absolute Shrinkage and Selection Operator, LASSO, Tibshirani,1996)的地方在于：不依赖模型假设而且也不是惩罚似然的思路。并且我们证明了在一定条件下这两种新的变量选择方法可以几乎以趋向于1的概率选对重要变量。
通过大量的模拟实验我们比较了本文中提出的各方法和一些已有的方法,进而验证了这些新方法的有效性。另外我们还将这些方法用于分析各种实际数据,比如棒球击打手的年薪数据,贝类生物horse mussel数据,淋巴癌数据以及波士顿房价数据,说明了这些方法的应用价值所在。
This dissertation is devoted to theoretical extensions and methodology development in the literature of sufficient dimension reduction.
There are two main focus in the literature of sufficient dimension reduction. The first topic is to estimate the basis directions of the Central (Mean) Subspace. However, the asymptotic property of the estimated directions based on classic methods is not very clear. To better understand the theoretical properties of existing sufficient dimension reduction methods especially for their estimated directions, we first derive the second order asymptotic expansions for their sample estimators of candidate matrix and basis directions. We take four most commonly used sufficient dimension reduction methods Sliced Inverse Regression (SIR, Li,1991), Sliced Average Variance Estimation (SAVE, Cook and Weisberg,1991), Principal Hessian Direction (PHD, Li,1992) and Directional Regression (DR, Li and Wang,2007) for illustration. As an application of the asymptotic results, the following task is to reduce the bias, a specific aim being removal of the leading bias of O(n-1). With the help of the second order asymptotic expansions of these methods, we can easily deduct the general formulae for their second order biases. Moreover, the second order bias for each method can be consistently estimated using the obvious sample analogues. Subtracting the sample estimators of the second order biases from the original estimators, we can construct the bias corrected estimators for these sufficient dimension reduction methods, which will be unbiased to O(n-1).
How to determine the structural dimension is another critical issue. Each classic method in determining structural dimension has its own shortcoming. Sequential tests relies heavily on the significance level. Bootstrap method is computationally intensive. Even Bayes information criterion proposed by Zhu, Miao and Peng (2006) can consistently estimate the structural dimension, the optimal form of its penalty function is difficult to identify in a data-driven manner. Moreover, traditional sufficient dimension reduction methods separate directions estimation and structural dimension determination into two steps. Our second contribution is to propose a sparse eigen-decomposition strategy by shrinking small sample eigenvalues to zero. Its main idea is to formulate the spectral decomposition of a kernel matrix into a least squares structure so that we can impose a penalty on sample eigenvalues. The adaptive Least Absolute Shrinkage and Selection Operator (Zou,2006) is recommended to produce sparse estimation of the eigenvalues so that we can estimate the structural dimension efficiently. Different from existing methods, the new method can simultaneously estimate basis directions and structural dimension of the Central (Mean) Subspace in a data-driven manner. The oracle property of our estimation procedure is also established.
The third part of this thesis is the B-spline approximation to the kernel matrix of classical sufficient dimension reduction methods SIR and SAVE. Compared with slicing method and kernel estimation that has been used in the literature, B-spline approximation is of higher accuracy and is easier to implement. In addition we suggest a a modified Bayes information criterion based on Zhu, Miao and Peng (2006) to estimate the structural dimension. The key idea here to choose the penalty function is to make the leading term and the penalty term comparable in magnitude. This modified criterion can help to enhance the efficacy of estimation.
Dimension reduction for semiparametric regressions involves two issues: recovering informative linear combinations of predictors and selecting contributing predictors. The first goal can be achieved by sufficient dimension reduction. The second goal is the current hot topic in statistics: variable selection. The fourth part of this thesis is to develop a new method to fulfill simultaneously sufficient dimension reduction and variable selection. Inspired by the ground breaking work of Candes and Tao (2007), We here suggest the (?-)1-regularization of sliced inverse regression by the Dantzig selector. The new proposal is designed to strike a balance between nearly solving the eigen-decomposition form of SIR and minimizing the (?)1 norm of the directions. Moreover, our new method can work efficiently even when p> n, where p is the dimension of predictors and n is the sample size. When the dimension p is fixed, the asymptotic properties (consistency in estimation and convergence in distribution) of the proposed estimators are investigated. Moreover, a bound on the estimation error is derived when both the sample size n and the dimension p tend to infinity, which allows p> n as a special case.
Finally we study how to test contributing predictors based on directional regression in a model free setting. We derive the asymptotic distribution of the test statistics proposed here under null hypothesis. Based on the asymptotics of the test statistics, we further propose two easy ways to implement variable selection. The new predictor selection proposals is different from any other hot variable selection methods like the Least Absolute Shrinkage and Selection Operator (LASSO, Tibshirani,1996) since it is model free and it does not involve any penalties. The two new approaches will correctly identify significant variable with probability tending to 1 under certain conditions.
Moreover, we compare each proposal in this dissertation with its related alternatives by comprehensive simulation studies to illustrate the efficiency of our proposals. We also demonstrate the use of our proposals through a wide range of applications in real data analysis, such as hitters'salary data, horse mussel data, lymphoma data and Boston housing data.
在充分降维领域中有两个重要的话题。第一个是估计中心(均值)降维空间的基方向。而经典降维方法基方向样本估计的大样本性质至今仍不明了。为了进一步了解这些常用降维方法特别是其方向估计的理论性质,在这篇论文中我们首先研究了这些方法的核矩阵以及基方向样本估计的二阶渐近性质。我们推导了四种常见的降维方法,包括切片逆回归(Sliced Inverse Regression, SIR, Li,1991),切片平均方差估计(Sliced Average Variance Estimation, Cook and Weisberg,1991),海赛主方向(Principal Hessian Direction, Li,1992)和方向回归(Directional Regression, Li and Wang,2007),它们样本估计的二阶渐近展开式。利用这些降维方法的二阶渐近展式,我们可以进一步考虑纠正其O(n-1)偏差以提高估计的精度。从已经得到的二阶渐近展式中,我们可以求出降维方法方向估计二阶偏差的显示表达式,继而可以很容易得到二阶偏差的相合样本估计。我们随后提出一种一般的降维方法的偏差纠正策略,其思想很简单：即是将某一种降维方法基方向的样本估计减去其二阶偏差的样本估计。并且我们证明了经过偏差纠正后,这些降维方法方向估计的偏差被缩小到O(n-2)。
充分降维领域中的另一个重要的问题是决定中心(均值)降维子空间的结构维数。常用的选取结构维数的方法都有其局限性。序贯检验法依赖于检验的显著性水平。重抽样方法的运算量过大。Zhu, Miao and Peng (2006)提出的BIC准则虽然可以相合的估计结构维数,但如何基于数据选取最优的惩罚函数是一个难题。更重要的是,一般充分降维的过程分为两步,首先是决定维数,然后再选取相应的基方向。论文的第二部分基于对降维方法核矩阵样本特征值的压缩估计提出一种稀疏谱分解方法用以决定结构维数。该方法的主要思路是通过建立矩阵谱分解与最小二乘之间的联系,然后利用Zou(2006)所提出的自适应性最小绝对缩减和变量选择算子得到样本特征值的稀疏估计。和以往降维方法的两步估计不同,稀疏谱分解方法可以同时估计结构维数和中心(均值)降维子空间的基方向。同时我们还证明了稀疏谱分解方法具有Oracle性质。
本文的第三部分是将非参数方法B样条用以估计降维方法SIR和SAVE的核矩阵。和已有的切片方法以及核估计方法相比,B样条方法估计精度更高并且也同样计算简单。另外我们修正了Zhu, Miao and Peng (2006)所提出的BIC准则。修正的目的是为了平衡BIC准则中的主项与惩罚项,使得其数量级大致相仿。这种修正的BIC准则在估计结构维数方面的精度较之传统方法也有所提高。
对于半参数模型的降维一般有两个出发点：一是找出模型中可能存在的变量的线性组合,二是选择模型中的重要变量。其中第一点就是充分降维的概念。而第二点是当前统计学界非常热门的话题：变量选择。本文的第四部分提出一种同时进行充分降维和变量选择的新方法。受到Candes and Tao(2007)一文的启发,我们借鉴DantzigSelector对SIR方法进行了e1规范化。这种新方法的本质是在从SIR的谱分解形式中求解基方向与极小化基方向e1模长之间的一种折中方案。我们所提出的这种新的充分降维与变量选择相结合的方法,在样本量n小于变量维数p的情形下仍然行之有效。当变量维数p固定时,我们得到了这种新方法的相合性和渐近正态性。当变量维数p和n以一定形式趋向于无穷时(需要指出的是p>n的情形只是这种形式的特殊情形),我们得到了这种方法估计的误差上界。
最后我们基于方向回归提出一种不基于模型假设的变量显著性检验。我们推导了在原假设下所提出的统计量的渐近分布。另外基于统计量的渐近性质,我们提出了两种非常简单的不基于模型假设的变量选择方法。这两种新方法不同于当前流行的变量选择方法比如最小绝对缩减和变量选择算子(Least Absolute Shrinkage and Selection Operator, LASSO, Tibshirani,1996)的地方在于：不依赖模型假设而且也不是惩罚似然的思路。并且我们证明了在一定条件下这两种新的变量选择方法可以几乎以趋向于1的概率选对重要变量。
通过大量的模拟实验我们比较了本文中提出的各方法和一些已有的方法,进而验证了这些新方法的有效性。另外我们还将这些方法用于分析各种实际数据,比如棒球击打手的年薪数据,贝类生物horse mussel数据,淋巴癌数据以及波士顿房价数据,说明了这些方法的应用价值所在。
This dissertation is devoted to theoretical extensions and methodology development in the literature of sufficient dimension reduction.
There are two main focus in the literature of sufficient dimension reduction. The first topic is to estimate the basis directions of the Central (Mean) Subspace. However, the asymptotic property of the estimated directions based on classic methods is not very clear. To better understand the theoretical properties of existing sufficient dimension reduction methods especially for their estimated directions, we first derive the second order asymptotic expansions for their sample estimators of candidate matrix and basis directions. We take four most commonly used sufficient dimension reduction methods Sliced Inverse Regression (SIR, Li,1991), Sliced Average Variance Estimation (SAVE, Cook and Weisberg,1991), Principal Hessian Direction (PHD, Li,1992) and Directional Regression (DR, Li and Wang,2007) for illustration. As an application of the asymptotic results, the following task is to reduce the bias, a specific aim being removal of the leading bias of O(n-1). With the help of the second order asymptotic expansions of these methods, we can easily deduct the general formulae for their second order biases. Moreover, the second order bias for each method can be consistently estimated using the obvious sample analogues. Subtracting the sample estimators of the second order biases from the original estimators, we can construct the bias corrected estimators for these sufficient dimension reduction methods, which will be unbiased to O(n-1).
How to determine the structural dimension is another critical issue. Each classic method in determining structural dimension has its own shortcoming. Sequential tests relies heavily on the significance level. Bootstrap method is computationally intensive. Even Bayes information criterion proposed by Zhu, Miao and Peng (2006) can consistently estimate the structural dimension, the optimal form of its penalty function is difficult to identify in a data-driven manner. Moreover, traditional sufficient dimension reduction methods separate directions estimation and structural dimension determination into two steps. Our second contribution is to propose a sparse eigen-decomposition strategy by shrinking small sample eigenvalues to zero. Its main idea is to formulate the spectral decomposition of a kernel matrix into a least squares structure so that we can impose a penalty on sample eigenvalues. The adaptive Least Absolute Shrinkage and Selection Operator (Zou,2006) is recommended to produce sparse estimation of the eigenvalues so that we can estimate the structural dimension efficiently. Different from existing methods, the new method can simultaneously estimate basis directions and structural dimension of the Central (Mean) Subspace in a data-driven manner. The oracle property of our estimation procedure is also established.
The third part of this thesis is the B-spline approximation to the kernel matrix of classical sufficient dimension reduction methods SIR and SAVE. Compared with slicing method and kernel estimation that has been used in the literature, B-spline approximation is of higher accuracy and is easier to implement. In addition we suggest a a modified Bayes information criterion based on Zhu, Miao and Peng (2006) to estimate the structural dimension. The key idea here to choose the penalty function is to make the leading term and the penalty term comparable in magnitude. This modified criterion can help to enhance the efficacy of estimation.
Dimension reduction for semiparametric regressions involves two issues: recovering informative linear combinations of predictors and selecting contributing predictors. The first goal can be achieved by sufficient dimension reduction. The second goal is the current hot topic in statistics: variable selection. The fourth part of this thesis is to develop a new method to fulfill simultaneously sufficient dimension reduction and variable selection. Inspired by the ground breaking work of Candes and Tao (2007), We here suggest the (?-)1-regularization of sliced inverse regression by the Dantzig selector. The new proposal is designed to strike a balance between nearly solving the eigen-decomposition form of SIR and minimizing the (?)1 norm of the directions. Moreover, our new method can work efficiently even when p> n, where p is the dimension of predictors and n is the sample size. When the dimension p is fixed, the asymptotic properties (consistency in estimation and convergence in distribution) of the proposed estimators are investigated. Moreover, a bound on the estimation error is derived when both the sample size n and the dimension p tend to infinity, which allows p> n as a special case.
Finally we study how to test contributing predictors based on directional regression in a model free setting. We derive the asymptotic distribution of the test statistics proposed here under null hypothesis. Based on the asymptotics of the test statistics, we further propose two easy ways to implement variable selection. The new predictor selection proposals is different from any other hot variable selection methods like the Least Absolute Shrinkage and Selection Operator (LASSO, Tibshirani,1996) since it is model free and it does not involve any penalties. The two new approaches will correctly identify significant variable with probability tending to 1 under certain conditions.
Moreover, we compare each proposal in this dissertation with its related alternatives by comprehensive simulation studies to illustrate the efficiency of our proposals. We also demonstrate the use of our proposals through a wide range of applications in real data analysis, such as hitters'salary data, horse mussel data, lymphoma data and Boston housing data.
引文
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