函数数据变换模型及降维方法的研究

项目名称： 函数数据变换模型及降维方法的研究

项目编号： No.11501248

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 王国长

作者单位： 暨南大学

项目金额： 18万元

中文摘要： 函数数据是一类新的数据类型，也是当前统计学界的研究热点。对函数数据建模研究的难点在于模型的高维和结构复杂。本项目主要研究两方面的内容: 函数变换模型和降维。针对现有方法只能应用于稠密的函数数据及预测仅含有函数曲线等缺陷。本项目主要研究两类新的变换模型：第一，研究函数预测曲线的观测是纵向数据（稀疏函数数据）变换模型；第二，是研究预测变量同时包含函数曲线和随机向量的变换模型。现有函数降维方法主要是 FSIR和FSAVE 及其变种。众所周知，FSIR和FSAVE 各自都有明显的优缺点。另外, 现有方法仅考虑含有函数预测的降维。针对现有方法的缺陷，本项目主要考虑两类新的函数降维：第一，考虑 对 FSIR和FSAVE 加权的降维方法；第二，考虑预测变量同时含有函数曲线和类别变量的降维方法。进一步, 本项目将研究新方法估计的收敛速度并用新方法来分析实际数据。

中文关键词： 函数数据；变换模型；降维；EDR；空间

英文摘要： Functional data is a new type of data, and is also a hot research area in current statistics. It is hard to model the functional data because of the functional data are high dimension and structural complexity. There are mainly two aspects of content for the project: functional transformation model and dimension reduction. There are two drawbacks for the existing functional transformation model methods which can only be applied to dense functional data and the predictors only contain functional curve. The projector mainly study two new transformation model: the first one is to study the transformation model when the observations of functional curve is longitudinal data (sparse functional data); the second one is to study when the predictors contain both functional curve and random vector. All of the existing functional dimension reduction methods are mainly the FSIR and FSAVE or its variety. It is well known that the merits and demerits for both FSIR and FSAVE are very clearly. Moreover, all of the existing methods only consider the dimension reduction methods which only consider the functional predictor. To aviod these drawbacks, the projector considers two new functioanl dimension reduction methods: the first one is that the project propose the weighted method for FSIR and FSAVE; the second one is to consider the dimension reduction method that the predictor contain both of functional curve and categorial variable. Furthermore, the projector will study the rate of convergence for the estimator of the proposed new methods and apply it to analyze the pratical data.

英文关键词： Functional data ;Transformaton model ;Dimension Reduction;EDR space