An Empirical Analysis Of A Nonlinear Functional Regression Model

With the continuous development of economy and society,the data field we face is becoming more and more complex and diversified,functional data as an emerging data is becoming more and more common in people’s daily life,playing an increasingly important role,how to deal with and make good use of such data processing problems in reality has become more and more people’s attention.Literature[2]proposes a new class of models in the study,nonlinear functional regression models(NFRM).Based on the literature[2]research,this paper applies its proposed nonlinear functional regression model(NFRM)to two types of data,Chinese and air pollution,in addition to the three different connection functions mentioned in the literature[2]:g(u)=cos(πu)g(u)=sin(πru)g(u)=u2/4+u/2In addition,this paper considers that the composite of the join function may have a better effect,so according to the relative magnitude of the nonparametric partial error and the overall error in the empirical analysis,the two join functions with small errors are selected:g(u)=cos(πu)g(u)=sin(πm)based on,give each of the two functions a weight of 1/2 to form a new connection function:g(u)=1/2 × cos(πu)+1/2 × sin(πu)Under the conditions of the above four connection functions,on the one hand,the nonparametric partial error MSE1 and the overall error MSE2 of the nonlinear functional regression model and the single-index functional model are compared horizontally and longitudinally to verify the feasibility and universality of the nonlinear functional regression model,and on the other hand,the connection function that fits the nonparametric part best for different data types is found in the case of nonlinear functional regression model.Regarding Chinese data,this paper selects five types of data from 1949 to 2021 as samples the total permanent population at the end of China,male,female,urban and rural permanent population data,and first functionalizes these five types of data,studies static data from a dynamic perspective,and then substitutes four connection functions for empirical analysis.Empirical studies show that for Chinese data,the connection function of the nonlinear functional regression model(NFRM)that fits the nonparametric part best is g(u)=u2/4+u/2.Comparing the nonlinear functional regression model with the unimetric regression model,it is found that when the connection function of the total national population,male population,female population and urban population is g(u)=cos(πu)or g(u)=u2/4+u/2,the fitting effect of the nonlinear functional regression model on the nonparametric part is improved,and the overall fitting effect is greatly improved.Through empirical research,it is found that for Chinese data,the connection function of the nonlinear functional regression model(NFRM)that fits the nonparametric part is g(u)=u2/4+u/2;comparing the nonlinear functional regression model with the unimetric regression model,the connection function of the total national population,male population,female population and urban population is g(u)=cos(πu)or g(u)=u2/4+ u/2.In rural population,when the connection function is g(u)=cos(πu)or g(u)=u2/4+u/2 or g(u)=1/2 × cos(πu)+1/2 × sin(πu),the fitting effect of the nonlinear functional regression model on the nonparametric part is improved,and the overall fitting effect is greatly improved.In this paper,the monthly average data of pollutant concentration values in Beijing in the past 110 months(December 2013-January 2023)were selected,and the research objects were pm2.5,pm10,NO2 and CO.Since the above data structure exhibits obvious functional characteristics over time,it is analyzed as a function curve.Through empirical analysis,it is found that the best-fitting connection function in the nonlinear functional regression model is g(u)=1/2 × cos(πu)+1/2 × sin(πu).Comparing the nonlinear functional regression model(NFRM)and the unimetric regression model,the fitting effect of the nonlinear functional regression model on the nonparametric part is improved more when the connection function is g(u)=cos(πu)or g(u)=u2/4+u/2 or g(u)=1/2 × cos(πu)+1/2 × sin(πu),and the fitting effect of the nonlinear functional regression model on the nonparametric part is improved,which plays a greater pulling effect on the overall fitting effect.