文章快速检索 高级检索

1. 北京航空航天大学经济管理学院, 北京 100083;
2. 城市运行应急保障模拟技术北京市重点实验室, 北京 100083;
3. 中央财经大学统计与数学学院, 北京 100081

Generalized linear regression model based on functional data analysis
WANG Huiwen1,2, HUANG Lele1,2, WANG Siyang3
1. School of Economics and Management, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. Beijing Key Laboratory of Emergency Support Simulation Technologies for City Operations, Beijing 100083, China;
3. School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China
Received: 2015-02-05; Accepted: 2015-03-05; Published online: 2015-03-30
Foundation items: National Natural Science Foundation of China (71420107025,11501586); National High-tech Research and Development Program of China (SS2014AA012303); 2014 Cultivation Project for Major Sciencific Research of Central University of Finance and Economics (Basic Theory)
Abstract: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.
Key words: functional data     generalized linear model     principal component     B spline     reweight

n个独立同分布的样本观测值分别为{Zi,Xi(t),Yi}i=1n。定义函数型数据X(t)的协方差函数和样本协方差函数分别为

2.2 〈X,β〉的处理

γ,B均为一维参数为例描述重加权算法。假设(γ(m),B(m))为在第m步迭代中得到的值。对对数似然函数求导可得

 误差水平 估计量 正态 Logistic 泊松 σ=0.2 γ1 0.5012 0.5138 0.5042 (0.0172) (0.1803) (0.0576) γ2 0.4999 0.5325 0.4910 (0.0168) (0.1662) (0.0556) σ=0.5 γ1 0.5013 0.4955 0.5297 (0.0364) (0.1961) (0.0870) γ2 0.5020 0.5399 0.5214 (0.0381) (0.1609) (0.0908) σ=1.0 γ1 0.5093 0.4313 0.6148 (0.0724) (0.1574) (0.1464) γ2 0.4959 0.4280 0.5696 (0.0663) (0.1618) (0.1586)

 样本容量 σ=0.2 σ=0.4 σ=0.6 100 0.1456 0.2145 1.1189 (0.1146) (0.1543) (0.8696) 200 0.0751 0.1180 0.7030 (0.0554) (0.0853) (0.6449) 300 0.0494 0.0826 0.6335 (0.0318) (0.0655) (0.6884) 500 0.0296 0.0410 0.4461 (0.0209) (0.0242) (0.5886)

 图 1 不同误差水平下对函数型系数的估计Fig. 1 Estimation for functional coefficient under different variances of error
5 结 论

1) 所提出的估计方法不需要对误差分布进行假设,扩大了适用范围。

2) 模型可以解决因变量为离散型或者属性数据的回归问题。

3) 将函数型数据分析方法引入了广义线性模型。

 [1] RAMSAY J O.When the data are functions[J].Psychometrika,1982,47(4):379-396. Click to display the text [2] MULLER H,WU Y,YAO F.Continuously additive models for nonlinear functional regression[J].Biometrika,2013,100(3):607-622. Click to display the text [3] DELSOL L,FERRATY F,VIEU P.Structural test in regression on functional variables[J].Journal of Multivariate Analysis,2011,102(3):422-447. Click to display the text [4] HE G,MULLER H,WANG J,et al.Functional linear regression via canonical analysis[J].Bernoulli,2010,16(3):705-729. Click to display the text [5] DELAIGLE A,HALL P.Classification using censored functional data[J].Journal of the American Statistical Association,2013,108(504):1269-1283. Click to display the text [6] HALL P,HOROWITZ J L.Methodology and convergence rates for functional linear regression[J].The Annals of Statistics,2007,35(1):70-91. Click to display the text [7] GHERIBALLAH A,LAKSACI A,SEKKAA S.Nonparametric M-regression for functional ergodic data[J].Statistics & Probability Letters,2013,83(3):902-908. Click to display the text [8] KATO K.Estimation in functional linear quantile regression[J].The Annals of Statistics,2012,40(6):3108-3136. Click to display the text [9] FERRATY F,GONZÁLEZ-MANTEIGA W,MARTÍNEZ-CALVO A,et al.Presmoothing in functional linear regression[J].Statistica Sinica,2012,22(1):69-94. Click to display the text [10] LIAN H.Shrinkage estimation and selection for multiple functional regression[J].Statistica Sinica,2013,23(1):51-74. Click to display the text [11] CANTONI E,RONCHETTI E.Robust inference for generalized linear models[J].Journal of the American Statistical Association,2001,96(455):1022-1030. Click to display the text [12] BOENTE G,HE X,ZHOU J.Robust estimates in generalized partially linear models[J].The Annals of Statistics,2006,34(6):2856-2878. Click to display the text [13] JAMES G M,WANG J,ZHU J.Functional linear regression that's interpretable[J].The Annals of Statistics,2009,37(5A):2083-2108. Click to display the text [14] CAMERON A C,TRIVEDI P K.Regression-based tests for overdispersion in the Poisson model[J].Journal of Econometrics,1990,46(3):347-364. Click to display the text [15] KIM M.Quantile regression with varying coefficients[J].The Annals of Statistics,2007,35(1):92-108. Click to display the text

#### 文章信息

WANG Huiwen, HUANG Lele, WANG Siyang

Generalized linear regression model based on functional data analysis

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(1): 8-12.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0078