A Functional Linear Model With AR Errors In Reproducing Kernel Hilbert Space And Its Applications

At present,the vigorous development of the era of big data makes big data fully enter the daily life of people,and the development of all works of life cannot be separated from the support of big data.Also,with the further development of the era of big data,the increasingly sophisticated modern data collection system changes the collected data from a static concept to a dynamic concept with real-time characteristics,that is,data exists widely in various fields in the form of function characteristics.This kind of special data is called functional data.Corresponding to the real research,functional data can be expressed as the time-varying ECG image information of different patients in the medical field,the real-time stock quotes in highfrequency trading,etc.Because of the special performance of functional data compared with ordinary data,the traditional data analysis method is no longer applicable,so the functional data analysis is further proposed.Today,the development of functional data has been more in-depth,and the commonly used functional data analysis tools mainly include functional principal component analysis,functional regression analysis,etc.However,most of the existing studies assume that the error terms are independent and identically distributed,and the dependent errors in the actual data are not deeply considered.Therefore,in combination with the current situation that the dependent structure of errors is seldom considered in the existing functional data analysis,this thesis proposes to describe the timing errors in the data by using the AR model in time series analysis.Based on the functional linear model,a parameter estimation method in the framework of Reproducing Kernel Hilbert Space--two-step estimation method is constructed.It is compared with the existing one-step estimation method without considering the special error structure.The specific contents of this thesis are as follows.Firstly,this thesis discusses the ubiquitous functional data in the context of existing big data research.By carding and integrating existing parameter estimation methods used in functional data analysis and the related questions of the main research in this field,this thesis considers the dependence error studies,which are relatively rare in existing functional data analysis.Therefore,based on the existing functional data analysis models,a functional linear model with AR errors is proposed.Secondly,based on the functional linear model with AR errors,the traditional parameter estimation method based on the assumption that the error term obeys the independent and identically distributed will no longer be effective.Therefore,this thesis proposes a parameter estimation method named two-step method under the framework of Reproducing Kernel Hilbert Space based on the existing research results,and the one-step estimation method without considering AR error is introduced.In order to verify the good performance of two-step estimation method in processing data with dependent errors,this thesis establishes simulation cases under different parameter settings to compare the estimation effects of one-step estimation method and two-step estimation method.Finally,considering the current problem of data dependence caused by the recurrent nature of diseases in the medical research field,this thesis obtained the hospitalization data of pediatric neurology department from a hospital in Changsha,Hunan Province,combined with the relationship between related diseases and temperature.A functional linear model with AR errors is constructed to describe the relationship between the number of hospitalizations for different diseases in a certain period and the daily temperature changes in the corresponding period,with daily temperature changes as the explanatory variable and the number of hospitalizations for febrile convulsion and viral encephalitis as the explanatory variables,which further verifies the good estimation effect of the two-step estimation method.In general,this thesis makes an in-depth study on the problem of dependence error,which has not been widely discussed in functional data analysis.A parameter estimation method based on a functional linear model with AR errors is proposed in the framework of Reproducing Kernel Hilbert Space,and the two-step method is compared with the one-step method without considering AR errors.Both simulation analysis and empirical analysis confirm that the twostep method has better estimation effect than the one-step method when there are dependent errors.In particular,the advantage of the two-step estimation method increases gradually with the increase of sample size.The problem considered in this thesis not only deepen the relevant research of functional data analysis,but also expand the application of the functional linear model,which is of great significance.