Measure of Dependence, Model Selection and Multiple Testing in Regression

本建议考虑多元回归程序来分析响应变量与协变量向量之间的关系。它引入了一种方法来处理这样的困境：对于高维数据，样本空间区域中数据的稀疏性使得通过选择相对变异性（噪声）最小的过程来估计非参数曲线和曲面几乎是不可能的。这是通过放弃试图估计真正的潜在曲线的目标，而是引入可能能够确定变量之间重要关系的依赖度量来实现的。这些依赖度量用调优常数来表示，调优常数是为了使信噪比最大化而选择的。更准确地说，调整参数是一个矢量，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参数的局部估计，噪声是该估计的标准误差（SE，标准差的估计）。这种选择窗口大小以最大化信噪比的方法解除了维度的诅咒，因为对于具有数据稀疏性的区域，SE非常大。它包括模型选择，其中与SE相比贡献不显著信号的变量被消除。在涉及大量因素和复杂关系的研究中，建议开发可用于确定因素之间关系的程序。建议的方法一般适用，没有任何限制条件。它涉及在研究中发现关键关系，其中有大量因素需要筛选其相关性。本提案中的降维和发现技术在各种科学和工程环境中都很有用，包括遗传学和生物信息学。总而言之，建议发展在涉及大量复杂数据集的研究中有用的方法，这些数据集在许多领域都很常见，包括卫生、环境和金融研究。

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.