Research On The Methods And Applications Of Functional Data Regression With Structural Information

Functional data analysis method analyzes problems from the perspective of the dynamic random process.It takes a changing curve track as a research unit to reflect the internal development law of things.In recent years,functional data analysis method has been developed rapidly.With the development of data collection technology and storage technology,we are facing more and more complex data structures.Therefore,how to use the structural information of data to improve the accuracy of regression model estimation and prediction is a problem worthy of study.This paper focuses on methods and applications of functional data regression with structural information and studies the models’ theoretical properties.The main contents are as follows:(1)We studied the functional linear model with prior information of sample networks.In many modern applications,samples come from individuals which are connected in a network.Network information plays an important role in forecasting.To improve the estimation effect and prediction effect,the proposed model considers prior information of sample network by the Laplace quadratic penalty function in the objective function.The statistical properties of the model are studied.With the development of data collection and storage technology,we are faced with more and more huge and complex data,which may contain multiple functional covariates.In view of this phenomenon,this paper used high-dimensional data processing technology to select variables for functional covariates,and proposed a high-dimensional functional linear model with the prior information of sample network.In view of the fact that there may be "outliers" in the real data analysis,this paper proposed a robust highdimensional functional linear model with the prior information of the sample network.The simulation results and case analysis results show that when there is network cohesion,the traditional model can be improved by incorporating the sample network structure information into the model prediction.(2)We studied subgroup analysis for high-dimensional functional linear model.In some cases,we only know that the data may come from different subgroups,but the specific sample source heterogeneity is unknown.Therefore,when building the model,a challenging problem is how to automatically group the samples and estimate the parameters when the heterogeneity information is unknown.In this paper,we proposed a high-dimensional functional linear model for heterogeneous high-dimensional functional data.The heterogeneity is determined by the unobserved potential factors,i.e.,the intercept term of regression.By punishing individual differences,the samples are automatically divided into subgroups.The model can automatically group the samples and select the variables from high-dimensional functional covariates at the same time.This is the first time to use subgroup analysis in functional data regression analysis.The statistical properties of the model are studied,and we allow the number of functional covariates and the number of sample heterogeneity groups to increase with the number of samples.Simulation analysis and example analysis show that the method has a good effect in dealing with heterogeneous data and homogeneous data.(3)We studied smooth and locally sparse estimation for multiple-output functional linear regression.In some practical data analysis,the same sample corresponds to multiple response variables,and the same covariate can build regression models for different response variables.Using the information of response variables to promote the estimation of multiple regression models is also a direction worthy of consideration.This paper studies how to use the relationship between different response variables to construct multiple regression models to improve the accuracy of local sparse(i.e.,zero on some sub-regions)estimation in each regression model.We propose a local sparse estimation method,i.e.,multiple smooth local sparse(m-SLoS)estimation.Simulation results and practical application show that the method has good numerical performance in estimating coefficient function,especially for the case that all multivariate response coefficient functions are the same.(4)We proposed multivariate functional generalized additive model.Multivariate functional data is a data set composed of multiple functions,widely in various disciplines and applications.In some cases,the relationship between response variables and functional covariates is not always linear.We removed the linear assumption under the sparse structure assumption.The high dimensional data analysis method is used to deal with this kind of problem,which is more flexible than the traditional functional linear model.When the link function is an identity function,the statistical properties of the model are studied.