A Nonparametric ANOVA Regression Model With Functional Inputs

In recent years,with the rapid development of digital economy,the data types available to various industries are becoming diversified,such as:Image type data,text type data.Functional data in industry,medicine,economics,biology and other fields has broad application prospects,so it also attracts a lot of scholars.Regression model is an important aspect in the research of functional data.Common functional regression models include scalar-on-functional model,functional-on-scalar model and functionalon-functional model.This paper studies functional-on-functional regression model.For this type of regression model,many non-parametric,semi-parametric or linear functional models and correlation estimation methods have been proposed.Nevertheless,there is no existing method for the situation where the functional covariates are bivariate functions with one of the variables in common with the response function.Aiming at this type of data,this paper proposes a non-parametric regression model with binary functional covariable and unary functional response variable,which can be widely applied to this kind of functional data obtained in various fields.In this paper,we first use the cubic spline kernel and the Gaussian kernel to construct the corresponding two reproducing kernel Hilbert Spaces,and build the model in the tensor product space of multiple reproducing kernel Hilbert Spaces.Secondly,the smoothing spline analysis of variance is used to decompose the reproducing Hilbert space and the non-parametric function in the model,and then the method combining functional empirical principal component and penalty least squares is used to estimate the non-parametric function in the model.Furthermore,for the constructed Gaussian kernel with functions as the variables,this paper also studies its properties as a reproducing kernel and the related properties of its corresponding reproducing kernel Hilbert space,and studies the convergence rate for the proposed estimation model method.Finally,in order to evaluate the fitting and prediction effect of the proposed model,data simulations are carried out,and two types of data are used in the case validation.One is about the movement data and CAHAI score data of stroke patients,and the other is about age-specific fertility data and mortality data.The feasibility of the proposed model and estimation method is verified by two groups of examples and the accuracy of the predicted results is verified by several simulation tests.