降维空间切片平均三阶矩估计的局部影响分析

发布时间：2018-05-22 10:07

  本文选题：局部影响分析 + 空间位移函数　； 参考：《云南财经大学》2017年硕士论文

【摘要】：在多元非参数回归问题中,有可能存在这样的情形:响应变量仅仅通过自变量的少数几个线性组合与自变量发生关联.在这样的情形下,找出这些自变量的线性组合可以降低回归的维数,从而带来一些回归结果的改善,比如:提高回归曲线拟合的精度、可视化,等等。各种充分降维方法的提出正是为了解决这个问题。在这些充分降维方法中,切片逆回归和切片平均方差估计方法是较为常用的两种方法。然而,当逆条件均值和逆条件方差为常量时,这两种方法均会失效。切片平均三阶矩估计方法的提出解决了这个问题并受到了广泛的关注。这种方法的使用需要估计自变量向量的条件三阶矩,所以,研究该方法的敏感性问题是有必要的。本文关注切片平均三阶矩估计法下中心子空间估计量的局部影响分析。本文在切片平均三阶矩估计法下提出的局部影响分析方法基于一个空间位移函数,该函数用于度量模型被扰动前后的中心子空间估计之间的差异。我们构建了一个切片平均三阶矩估计法局部影响分析的基本理论框架,这个框架下的所有关键量(如:拟曲率和强影响方向)的表达式都可以获得。在此框架下,局部影响评价统计量——最强影响方向,可以通过最小化拟曲率轻易地获取,因为后者可以表示为扰动方向的一个二次型。因此,这个方法的计算负担较轻。为了评价各个样本点对中心子空间估计的影响,我们设计了一个扰动方案,并在这个扰动方案下推导出了拟曲率和最强影响方向的具体表达式。为了说明本文所提出的上述方法,我们将其应用于一组模拟数据,该数据从一个经典的模型中产生,该模型中自变量向量的逆条件均值和逆条件方差均为常量。在这个模型下,切片平均三阶矩估计表现良好,而切片逆回归和切片平均方差估计方法均失效。模拟结果显示,本文提出的局部影响分析方法可以成功地识别出人为设置的异常点,此外,模拟结果还展示出了一些有趣的新发现。
[Abstract]:In multivariate nonparametric regression problems, it is possible that response variables are associated with independent variables only through a few linear combinations of independent variables. In this case, finding out the linear combination of these independent variables can reduce the dimension of regression and bring about some improvement of regression results, such as improving the precision of regression curve fitting, visualization, and so on. All kinds of sufficient dimensionality reduction methods are proposed to solve this problem. Among these sufficient dimensionality reduction methods, slice inverse regression and slice mean variance estimation are two common methods. However, when the inverse conditional mean and inverse conditional variance are constant, both methods will fail. The method of slice average third order moment estimation solves this problem and is paid more and more attention. The use of this method requires the estimation of conditional third-order moments of independent variable vectors, so it is necessary to study the sensitivity of the method. This paper focuses on the analysis of the local influence of the central subspace estimator under the third order moment estimation of slice average. In this paper, the local impact analysis method based on slice average third-order moment estimation is based on a spatial displacement function, which is used to measure the difference between the central subspace estimates before and after the model is disturbed. We construct a basic theoretical framework for local impact analysis of slice average third-order moment estimation, in which expressions of all key quantities (such as quasi curvature and strong influence direction) can be obtained. In this framework, the local impact evaluation statistics-the strongest direction of influence-can be easily obtained by minimizing quasi curvature, which can be expressed as a quadratic form of the direction of disturbance. Therefore, the computational burden of this method is relatively light. In order to evaluate the influence of each sample point on the estimation of the central subspace, we design a perturbation scheme and derive the specific expressions of the quasi curvature and the direction of the strongest influence under the perturbation scheme. In order to illustrate the above method proposed in this paper we apply it to a set of simulation data which is generated from a classical model in which the inverse conditional mean and inverse conditional variance of the independent variable vector are constant. In this model, slice average third-order moment estimation is good, while slice inverse regression and slice average variance estimation are invalid. The simulation results show that the proposed local impact analysis method can successfully identify the artificial outliers. In addition, the simulation results also show some interesting new findings.
【学位授予单位】：云南财经大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1
，

本文编号：1921626