Research Of Nonlinear Functional Regression Model

In the era of rapid progression of big data technology,the forms of collecting and storing data are also constantly changing.And a variety of complex types of data sets have emerged in various industries.Functional data is one of the complex data.Functional data has comprehensive application prospects,and its main research direction is functional regression models.At present,many scholars have proposed a large number of functional regression models,for instance,single-index functional regression models,multi-index functional regression models,partial linear functional regression models,partial single-index functional regression models,and so on.Based on this research,we propose a new functional regression model to analyze functional data in this thesis,which is a nonlinear functional regression model.In this thesis,we combine Simpson’s method with the B-spine method.First of all,we use Simpson’s method to solve the definite integral part of the nonlinear functional regression model.Secondly,the nonparametric connection function in the nonlinear functional regression model is approximated by using the cubic B-spline method.Finally,based on the above steps,a loss function is constructed by using the least-squares method,and the unknown parameters of the nonlinear functional regression model is estimated.At the same time,under certain assumptions,we not only present the convergence rate of the nonparametric partial estimates,but also give the consistency of the variance estimates.In this thesis,we conduct a numerical simulation study on the nonlinear functional regression model and compare it with the single-index functional model.According to the simulation results of the model,we get the following three conclusions:firstly,when the sample size increases continuously,regarding the nonlinear functional regression model and the single-index functional model,the estimated mean and variance of the non-parametric part show a downward trend,the estimated mean and variance of the whole part also show a downward trend;secondly,under different sample sizes,the mean of the non-parametric part and the overall part of the nonlinear functional regression model is smaller than the mean of the non-parametric part and the overall part of the single-index functional model;thirdly,according to the fitting effect of the model,the figure shows that the fitting effect of the nonparametric connection function of the model is getting better and better with the continuous increase of the sample size.Therefore,through research on numerical simulation,it is shown that the proposed new model can solve the regression problem of nonlinear functional data and the estimation accuracy is high.