高维数据中的条件非线性相关研究

本课题的研究对象是高维数据中的条件非线性相关，本质上是研究当一些协变量u给定时，自变量x与因变量Y之间的非线性相关系数。事实上，在许多情况下，我们需要根据新获得的信息更新自变量与因变量之间的关系，而且我们有时候只是对因变量的某个特定部分感兴趣，比如均值、分位数或者某个分位数区间。具体来说：.(1) 提出一种既能够避免维数灾难的问题，又不依赖于别的参数的选择的方法度量高维协变量x与因变量Y之间的非线性相关性；.(2) 计算因变量关于高维条件变量的中心子空间，并提出一种新的度量条件独立性的指标，研究该指标估计值的大样本性质；.(3) 创新性的提出条件区间独立性的概念，将条件分位数独立与条件独立联系起来。给出一种度量区间独立性的指标，并研究该指标估计值在原假设和备择假设下的渐近性质。

This study is concerned with conditional nonlinear dependence in high-dimensional data, which is essentially to study the nonlinear correlation between covariate x and response variable Y when the covariate u is given. In fact, in many cases, we need to update the relationship between the covariate x and response variable based on newly obtained information, and sometimes we are only interested in a particular part of the response variable, such as mean, quantile, or a quantile interval. Specifically:.(1) We propose a method that can avoid the dimensionality curse without relying on the choice of tuning parameters to measure the nonlinear dependence between high-dimensional covariate x and response variable Y;.(2) We calculate the center subspace of the response variable conditional on the high-dimensional covariate and propose a new index to measure the conditional independence. We also study the large sample property of the estimated value of the index;.(3) We innovatively put forward the concept of conditional interval quantile independence, which connects the conditional quantile independence with conditional independence. We present an index to measure the conditional interval quantile dependence and study the asymptotic properties of the estimated index under both the null hypothesis and the alternative hypothesis.