几类混合函数型数据建模
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摘要
随着测量工具的进步、收据收集和存储能力的提升,在许多应用领域(如化学计量学,生物计量学,医学,经济计量学等),我们能够在越来越精细的分辨率下收集、处理数据.例如,为了了解一天的温度变化曲线,我们可以每分钟记录一个数据,从而得到近似一天的温度变化曲线.对于这种数据,传统的统计模型面临极大的挑战,如过拟合和维数祸根问题.为克服这种困难,统计学者们提出了函数型数据分析方法,也即,将对每个个体的观测看成一条曲线,从而对曲线数据进行统计分析.回归分析一直是统计分析中一个重要的分析方法,它被许多科研工作者广泛的研究.最近,一些统计学者将传统的回归模型推广到函数型数据情况下,如Ramsay和Sliverman([81],[83])在其著作中研究了各种函数线性回归模型,Ferraty和Vieu [52]考虑了非参数函数型模型.此外,为了提高函数型回归模型的解释和预测能力,有些统计学者引入了额外的随机变量,我们称之为混合函数型数据回归模型.
本论文主要研究混合函数型数据回归建模问题,并考虑用多项式样条估计和惩罚样条估计两种非参数估计方法来对模型进行估计.
在第二章,我们引入了半函数线性模型.在该模型下,我们研究了多项式样条估计.在一定的正则条件下我们得到了估计的全局收敛速度和一致收敛速度.通过模拟实验我们研究了估计的有限样本性质,同时与半函数部分线性模型[3]和部分函数线性模型[96]进行了比较,说明我们的模型的可行性.
第三章,我们研究了部分函数线性模型[96]的多项式样条估计.在一定的正则条件下,我们同样得到了参数估计的渐近正态性和函数系数估计的全局收敛速度.通过模拟实验,我们研究了估计的有限样本性质,并跟文献[96]中的方法进行了比较,说明了我们的方法的优良性.
第四章,由于我们发现部分函数线性模型[96]的多项式样条估计不够稳健,从而提出了一个更加稳健的估计-惩罚样条估计.同样地,在一定的正则条件下,我们得到了参数部分估计的渐近正态性和函数系数估计的全局收敛速度.通过模拟实验,我们研究了惩罚样条估计的有限样本性质,同时通过比较,说明了惩罚样条估计在三种估计方法中是最好的.
第五章,我们引入了一个新的混合函数型数据模型-变系数部分函数线性模型.我们研究了该模型的多项式样条估计.在给定的正则条件下,我们研究了估计的全局收敛速度和一致收敛速度.通过模拟实验,研究了多项式样条估计的有限样本性质.
With the advances of measuring instruments and the improvement of data collect-ing and storage capacity, in many fields of applied sciences (such as chemometrics, biometrics, medicine, econometrics, etc.), we can collect and deal with the observa-tion data at finer and finer resolution. For example, For example, in order to under-stand the daily temperature curve, we can write down the temperature in every minute. And then, we can get an approximate daily the temperature curve. For this kind of data, the traditional statistical models to face great challenges, such as over-fitting and dimension curse problem. To overcome this difficulty, statistics have put forward functional data analysis methods. That is, each individual observation will be seen as a curve. Then, we can do statistical analysis based on the curve data. Regression analysis is aways an important analytical methods in the statistical analysis. It has been widely studied by lots of scientists. Recently, some statisticians extended the traditional regression model to the functional data case. For instance, Ramsay and Sliverman,([81],[83]) Studied the various functional linear regression models in their monograph. Ferraty and Vieu [52] considered the nonparametric functional regression model. In addition, in order to improve the power of interpretation and prediction of regression model, some statisticians introduced some additional random variables in the functional regression model, which is called the mixed-functional data regression model.
In this thesis, we study the problem of mixed functional data modeling and con-sider two kinds of nonparametric methods:polynomial spline method and penalized spline method.
In chapter2, we introduced the semi-functional linear model. In this model, we studied the polynomial spline estimation. Under some regularity conditions, we got the global and uniform convergence rates. By simulation study, we considered the finite-sample properties of the proposed estimator. We also compared our model with semi-functional partial linear regression [3] and partial functional linear model [96], which showed the feasibility of our model.
In chapter3, we considered the polynomial spline estimation of partial functional linear model [96]. Under some regularity conditions, we also got the asymptotic nor-mality on the parameter estimators and the global convergence rate of functional co-efficient. The finite-sample properties of the proposed estimators have been examined by a simulation. Compared with the estimation method of [96], the results showed our method is superior to [96].
In chapter4, we put forward a more robust estimation method for partial func-tional linear model-penalized spline estimation. Similarly, under some regularity con-ditions, we also got the asymptotic normality on the parameter estimators and the global convergence rate of functional coefficient. We studied the finite-sample prop-erties of penalized spline estimation by a simulation. Meanwhile, we compared the three methods of estimation, the results showed the penalized spline estimation is the best of all.
In chapter5, we introduced a new mixed functional data modeling-varying coef-ficient partially functional linear model. We studied the polynomial spline estimators of this model. Under the given conditions, we obtained the global and uniform con-vergence rates. By simulation experiment, we considered the finite-sample properties of the proposed estimator.
本论文主要研究混合函数型数据回归建模问题,并考虑用多项式样条估计和惩罚样条估计两种非参数估计方法来对模型进行估计.
在第二章,我们引入了半函数线性模型.在该模型下,我们研究了多项式样条估计.在一定的正则条件下我们得到了估计的全局收敛速度和一致收敛速度.通过模拟实验我们研究了估计的有限样本性质,同时与半函数部分线性模型[3]和部分函数线性模型[96]进行了比较,说明我们的模型的可行性.
第三章,我们研究了部分函数线性模型[96]的多项式样条估计.在一定的正则条件下,我们同样得到了参数估计的渐近正态性和函数系数估计的全局收敛速度.通过模拟实验,我们研究了估计的有限样本性质,并跟文献[96]中的方法进行了比较,说明了我们的方法的优良性.
第四章,由于我们发现部分函数线性模型[96]的多项式样条估计不够稳健,从而提出了一个更加稳健的估计-惩罚样条估计.同样地,在一定的正则条件下,我们得到了参数部分估计的渐近正态性和函数系数估计的全局收敛速度.通过模拟实验,我们研究了惩罚样条估计的有限样本性质,同时通过比较,说明了惩罚样条估计在三种估计方法中是最好的.
第五章,我们引入了一个新的混合函数型数据模型-变系数部分函数线性模型.我们研究了该模型的多项式样条估计.在给定的正则条件下,我们研究了估计的全局收敛速度和一致收敛速度.通过模拟实验,研究了多项式样条估计的有限样本性质.
With the advances of measuring instruments and the improvement of data collect-ing and storage capacity, in many fields of applied sciences (such as chemometrics, biometrics, medicine, econometrics, etc.), we can collect and deal with the observa-tion data at finer and finer resolution. For example, For example, in order to under-stand the daily temperature curve, we can write down the temperature in every minute. And then, we can get an approximate daily the temperature curve. For this kind of data, the traditional statistical models to face great challenges, such as over-fitting and dimension curse problem. To overcome this difficulty, statistics have put forward functional data analysis methods. That is, each individual observation will be seen as a curve. Then, we can do statistical analysis based on the curve data. Regression analysis is aways an important analytical methods in the statistical analysis. It has been widely studied by lots of scientists. Recently, some statisticians extended the traditional regression model to the functional data case. For instance, Ramsay and Sliverman,([81],[83]) Studied the various functional linear regression models in their monograph. Ferraty and Vieu [52] considered the nonparametric functional regression model. In addition, in order to improve the power of interpretation and prediction of regression model, some statisticians introduced some additional random variables in the functional regression model, which is called the mixed-functional data regression model.
In this thesis, we study the problem of mixed functional data modeling and con-sider two kinds of nonparametric methods:polynomial spline method and penalized spline method.
In chapter2, we introduced the semi-functional linear model. In this model, we studied the polynomial spline estimation. Under some regularity conditions, we got the global and uniform convergence rates. By simulation study, we considered the finite-sample properties of the proposed estimator. We also compared our model with semi-functional partial linear regression [3] and partial functional linear model [96], which showed the feasibility of our model.
In chapter3, we considered the polynomial spline estimation of partial functional linear model [96]. Under some regularity conditions, we also got the asymptotic nor-mality on the parameter estimators and the global convergence rate of functional co-efficient. The finite-sample properties of the proposed estimators have been examined by a simulation. Compared with the estimation method of [96], the results showed our method is superior to [96].
In chapter4, we put forward a more robust estimation method for partial func-tional linear model-penalized spline estimation. Similarly, under some regularity con-ditions, we also got the asymptotic normality on the parameter estimators and the global convergence rate of functional coefficient. We studied the finite-sample prop-erties of penalized spline estimation by a simulation. Meanwhile, we compared the three methods of estimation, the results showed the penalized spline estimation is the best of all.
In chapter5, we introduced a new mixed functional data modeling-varying coef-ficient partially functional linear model. We studied the polynomial spline estimators of this model. Under the given conditions, we obtained the global and uniform con-vergence rates. By simulation experiment, we considered the finite-sample properties of the proposed estimator.
引文
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