基于函数型数据的广义线性回归模型

函数型数据的回归分析研究主要集中在函数型线性模型。不要求因变量为连续型随机变量,可以为离散型或属性数据(对应于泊松或Logistic回归),对同时含有数值型多元变量和函数型变量的广义线性模型的估计问题进行分析,采用非参数方法得到了参数部分和非参数部分的估计量,并给出了一种重加权算法进行参数求解,解决了含数值型和函数型混合数据类型自变量的回归问题,同时扩展了函数型线性模型的应用范围。估计过程中,分别采用了函数型主成分和B样条基函数,并给出了基函数个数选择的准则。数值模拟结果表明,所提出方法具有良好的可行性与正确性。 

Generalized linear regression model based on functional data analysis

Functional linear regression model has captured much attention in functional data analysis. By tools in semiparametric and nonparametric statistics, it is proposed to estimate the coefficients in generalized linear regression models with both multivariate scalar covariates and functional covariates. In this framework, the theory of generalized linear model is introduced, and the response variable is not required to be continuous random variable and may be discrete or attribute data, which widely broadens the application of functional linear model by solving the regression problem of predictors with mixed types of multivariate data and functional data. Besides, Logistic regression and Possion regression corresponding to categorical or discrete responses were emphasized, and a reweight algorithm for maximizing the log likelihood function was provided. In the procedure of estimation, functional principal component analysis and B spline were utilized, and the criterion to select the number of basis functions was suggested. The simulation results show that the proposed estimation and test methods are effective.