Measure of Dependence, Model Selection and Multiple Testing in Regression

该建议考虑了用于分析响应变量和协变量向量之间关系的多元回归程序。 它介绍了一种方法，处理的困境，高维数据的样本空间的区域中的数据的稀疏性，使估计的非参数曲线和曲面几乎不可能通过选择程序的相对变异性（噪声）最小化。 这是通过放弃试图估计真实的潜在曲线的目标，而是引入可能能够确定变量之间的重要关系的相关性度量来实现的。 这些依赖措施表示的调谐常数，areschose最大化的信号噪声比。 更准确地说，调整参数是一个向量，它给出了协变量空间中局部区域的窗口大小，在协变量空间中，我们对响应变量进行局部参数拟合。 信号是依赖于窗口大小的依赖性参数的局部估计，并且噪声是该估计的标准误差（SE，标准偏差的估计）。 这种选择窗口大小以最大化信噪比的方法解除了维度灾难，因为对于具有稀疏数据的区域，SE非常大。它包括模型选择，其中的变量，贡献无关紧要的信号相比，他们的SE的被eliminated.It建议开发程序，可用于确定在研究中涉及大量的因素和复杂的关系因素之间的关系。所提出的方法是普遍适用的，没有任何限制条件。它涉及在研究中发现关键关系，其中有大量的因素需要筛选其相关性。降维和发现技术在这个建议是有用的，在各种科学和工程背景下，包括遗传学和生物信息学。总之，建议开发的方法，将是有用的研究，涉及大型和复杂的数据集，是常见的许多领域，包括健康，环境和金融的研究。

This proposal considers multiple regression procedures for analyzing the relationship between a response variable and a vector of covariates. It introduces an approach which deals with the dilemma that with highdimensional data the sparsity of data in regions of the sample space makes estimation of nonparametric curves and surfaces virtually impossible by choosing procedures whose relative variability (noise) is minimized. This is accomplished by abandoning the goal of trying to estimate true underlying curves and instead introducing measures of dependence that may be able to determine important relationships between variables. These dependence measures are expressed in terms of tuning constants that arechosen to maximize a signal to noise ratio. More precisely, the tuning parameter is a vector which gives the window size of local regions in the covariate space where we do local parametric fits of the response variableto the covariates. The signal is a local estimate of a dependence parameter which depends on the window size, and the noise is the standard error (SE, an estimate of the standard deviation) of this estimate. This approach of choosing the window size to maximize a signal to noise ratio lifts the curse of dimensionality because for regions with sparsity of data the SE is very large. It includes model selection where the variables that contribute insignificant signals compared to their SE's are eliminated.It is proposed to develop procedures that can be used to determine relationships between factors in studies involving a large number of factors and complex relationships. The proposed methodology is applicablegenerally without any restrictive conditions. It involves the discovery of key relationships in studies where there are a great number of factors that need to be screened for their relevance. The dimension reduction anddiscovery techniques in this proposal are useful in a variety of scientific and engineering contexts including genetics and bioinformatics. In summary, it is proposed to develop methods that will be useful in studies involving large and complex data sets that are common in many areas including studies of health, the environment, and finance.