图像分析中针对函数型响应变量的变系数模型

发布时间：2018-02-03 13:31

本文关键词： 图像分析函数型响应变量变系数模型全局检验统计量同时置信带弱收敛一致收敛　出处：《华东师范大学》2016年博士论文　论文类型：学位论文

【摘要】：近年来,在许多图像分析中,大量的函数型据被广泛地收集,这些函数型数据通常对应着某个空间中的一组网格点。人们希望将函数型数据与各种临床变量,比如年龄和性别等等联系起来,以解决科学实验中大家所感兴趣的问题。最近有许多神经影像学文献对图像数据做了大量的分析,受此启发,我们在这篇论文中提出三个不同的变系数模型,来研究图像分析中的函数型数据与一组协变量之间的关联。在第二章中,我们提出了非线性变系数混合效应模型(nonlinear varying mixed effects models, NVMEM),来研究纵向函数型响应变量和协变量之间的关系,它比现有的线性方法更灵活。我们提出了有效的估计方法和一些统计推断过程,并且系统地研究了它们的理论性质。我们分析了固定效应函数的估计,及其渐近偏差和渐进方差的弱收敛性质。我们探讨了协方差算子及其相关的特征值和特征函数估计的一致收敛速度,然后推导了随机效应函数估计的渐近偏差和积分均方误差,并计算了它们的一致收敛速度。我们为固定效应函数的线性假设提出了一个全局检验统计量,并得到了它在零假设下的渐进分布。我们还为固定效应函数构造了同时置信带。我们通过蒙特卡罗模拟研究了NVMEM在有限样本下的表现,并且用它分析了一个自闭症研究数据。在第三章中,我们提出了转换变系数模型(transformation varying coefficient model, TVCM),来研究函数型数据和一组协变量之间的关系。在分析响应变量和协变量的传统回归模型中,线性形式是最常用。然而在实际情况中,这一线性假设往往无法满足。我们提出了转换变系数模型的估计方法和统计推理过程,来研究这种非线性关系。我们还为变系数函数构建了同时置信带,并且系统地讨论了各种估计的理论性质。我们通过模拟研究评估了TVCM在有限样本下的表现,并且做了一个实际图像数据的应用。在第四章中,我们提出了单指标变系数模型(single-index varying coefficient model, SIVCM),来研究图像分析中的函数型数据与一组协变量之间的关联。SIVCM结合了变系数模型和单指标模型的几个特有的优点。我们提出了变系数函数、指标函数和个体协方差函数的估计方法。我们描述了如何将不同网格点上的信息最优化地整合起来,并且系统地研究了估计的渐近性质(例如相合性和收敛速度)。我们还为变系数函数构造了同时置信带。我们通过模拟研究评估了SIVCM在有限样本下的表现,并且用一个阿尔茨海默氏病疾病数据的研究证实了SIVCM具有比其它变系数模型更高的准确性。
[Abstract]:In recent years, in many image analysis, a large number of functional data are widely collected, these functional data usually correspond to a set of grid points in a certain space. Age and sex, for example, have been linked to solve problems of interest in scientific experiments, which have been inspired by the recent extensive analysis of image data in many neuroimaging literature. In this paper, we propose three different variable coefficient models to study the correlation between functional data and a set of covariables in image analysis. A nonlinear varying mixed effects model (NVMEM) is proposed. It is more flexible than the existing linear method to study the relationship between the longitudinal functional response variables and the covariables. We propose an effective estimation method and some statistical inference processes. And the theoretical properties of them are studied systematically, and the estimation of fixed effect function is analyzed. We discuss the uniform convergence rate of covariance operators and their associated eigenvalues and eigenfunction estimates. Then the asymptotic deviation and integral mean square error of random effect function estimation are derived and their uniform convergence rate is calculated. A global test statistic is proposed for the linear hypothesis of fixed effect function. We also construct simultaneous confidence bands for fixed effect functions. We study the behavior of NVMEM in finite samples by Monte Carlo simulation. And used it to analyze an autism study. In Chapter 3. We propose a transformation varying coefficient model (TVCM). To study the relationship between functional data and a set of covariables. In the traditional regression model of analyzing response variables and covariables, the linear form is the most commonly used. However, in the actual situation. This linear assumption is often not satisfied. We propose the estimation method and statistical reasoning process to study the nonlinear relationship. We also construct a simultaneous confidence band for the variable coefficient function. And we systematically discuss the theoretical properties of various estimates. We evaluate the performance of TVCM under finite samples through simulation research, and make an application of actual image data. In Chapter 4th. We propose a single-index varying coefficient model (SIVCM). To study the correlation between functional data and a set of covariables in image analysis, SIVCM combines several unique advantages of variable coefficient model and single parameter model. We propose variable coefficient function. We describe how to optimize the integration of information on different grid points. Furthermore, the asymptotic properties of the estimator (for example, consistency and convergence rate) are studied systematically. We also construct a simultaneous confidence band for variable coefficient functions. We evaluate the performance of SIVCM in finite samples through simulation studies. And a study using Alzheimer's disease data confirms that SIVCM is more accurate than other variable-coefficient models.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O212.1

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