Function-on-Partially Linear Functional Additive Models

We consider a functional partially linear additive model that predicts a functional response by a scalar predictor and functional predictors. The B-spline and eigenbasis least squares estimator for both the parametric and the nonparametric components proposed. In the final of this paper, as a result, we got the variance decomposition of the model and establish the asymptotic convergence rate for estimator.


Introduction
Function data are infinite dimensional vectors in functional space, the study of functional data helps people to further understand data changes in finance, medicine, etc. The study of fucntional response can date back to the work Ramsay and Silverman (2005) [1] where the model the functional concurrent model considers current response relate to the current values of the covariates. Wong et al. (2018) [2] investigated a class of partially linear functional additive models (PLFAM) that predicts a scalar response by both parametric effects of a multivariate predictor and nonparametric effects of a multivariate functional predictor.
Further an additive function-on-function regression is established using principal component basis functions and b-spline basis functions by Janet S. Kim et al. (2016) [3]. Luo, R., Qi, X. (2017) [4] consider functional linear regression models with a functional response and multiple functional predictors, with the goal of finding the best finite-dimensional approximation to the signal part of the response function. This paper extends function on function regression, con-( ) ( ) ( ) ( ) ( ) ( ) 1 , d s t x s s t τ µ γ β ε = = + + ∑ ∫ (1) where ( ) Y t is the response function defined in an interval y τ , For convenience, we assume that the response has zero-mean, the predictive curves.
( ) ( ) ( ) 1 2 , , , p x s x s x s are defined in x τ . where y τ and x τ are compact in- is the noise function with mean zero and unknown autocovariance function ( ) , A t t′ and is independent of the predictor and the response. In the previous papers, these focused on function-on-function, scalar-on-function, and scalar-mixed data. There is no paper devoted to function-on-mixed data. That is to say, the functional dependent variable analyzes the regression model of the mixed data. This paper considers the actual situation is closer to the actual demand. Considering that the real world data cannot be all the functional data, there will definitely be a part of the scalar in the model. From this perspective, expand the new model. This paper uses a different estimation method from the previous. The general method adopted by these articles is to use non-parametric methods such as spline function to approximate the model generation estimation parameters, and at the same time minimize the objective function. In the past, the model was generally complicated and inconvenient to calculate. They consider fitting with B-splines in three directions at the same time, and there will be over-fitting when doing so. In this paper, the problem is optimized by using the eigenbasis and orthogonal B-spline, that is, using the special feature base of ( ) Y t and an orthogonal B-spline to fit the model. While ensuring the fitting effect, the complexity of the model is reduced, and the model is not over-fitting.
We use the classical formula of probability to express the error term of the model from the perspective of variance, and the method of resampling is used to approximate the error of the model. At the same time, assume from the article hypothesis, we establish the asymptotic convergence rate for estimator.

Model
In this section, we consider the estimation of coefficients in a partial linear regression model of a function. In this paper, the model is estimated by using the eigenbasis spline and the B-spline function. Note k φ is the corresponding eigenfunctions of ( ) y t , and we project ( ) h t onto the eigenbasis of ( ) then we can express ( ) For the function coefficient term part of (1), we estimate it by means of Bspline and feature base. For convenience we write model (1) as In this paper, on spectral decomposition method projecting the corresponding function to the orthogonal eigenbasis of y, we estimate the parameter of the model can be performed without punished the complexity of the t direction.
Generally, we preset the maximum variance number similar to the method in the functional principal component analysis, usually 85%. Next we will use the Orthogonal b-spline method (2010) [5] [6] to represent the parameter items in the s direction.
In the t direction perform truncation K for the preset maximum variance, we We denote ( ) , , , Then we can get the simplified expression of ( ) y t Journal of Applied Mathematics and Physics

Parameter Estimation
We assume that with probability 1, the trajectory of n x is contained in a Hilbert space n χ , with inner product. We will focus on the case that j χ 's are 2 L functional spaces and the inner products are We use quadratic penalties for the direction s and control the roughness in the direction t by the preset number of orthogonal basis functions. so the loss function is as follow We use the least squares penalty to punish the curvature of the functional parameter, considering the penalty term where ( ) where t e is zero-mean error, and the variables k ξ are the FPCA (Functional principal component) scores of Y. And note that , so the criterion can be simplify written as So the estimation process for the parameters k H , k Φ is as follows.

Error Variance Decomposition
In this section, we get error variance decomposition. Let ( ) , A t t′ be the variance function of ( ) In this paper, we assume that π is a known parameter set containing estimated arbitrary parameter values and corresponding parameter variance and covariance functions. So we will expand  . This paper hopes to obtain an approximation form of the finite term of the variance function under finite samples by resampling the bootstrap method. We perform the bootstrap algorithm through the following steps; 2) Resample the samples to get Q groups different sample.

Basic Assumptions and Convergence Properties
In this section,we study the asymptotic properties of

Related Lemma
it is known by Chen [8] that

Convergence Properties
Consider the model (6) Use the principal component of ( ) y t for the model (6) to represent the following expression We can get an estimation by least squares estimation of Assume from the article hypothesis, we have ☐ Theorem 3. Available under the conditions of (6)-(9), and let Proof.
Consider the latter item in the text, because Now consider the former item ( ) ( ) ( ) ( ) Then consider

Conclusion
This paper is a model extension of the function-to-function regression. The scalar variable is added to the dependent variable, which extends the application scope of the model. For real-life data, real data should include scalar data and function data, and the model used in this paper can be better explained. In this paper, the model is estimated by using the eigenbasis spline function and the orthogonal B-spline function. When the loss function is punished, the complexity of the t direction is controlled by controlling the preset principal component in the t direction, which reduces the complexity. It is more practical. At the same time, the variance decomposition of the error under the finite sample size is given. The approximation is performed by the resampling bootstrap method.
Finally, the convergence properties of the estimated parameters are studied.