Functional regression

Functional regression is a version of regression analysis when responses or covariates include functional data. Functional regression models can be classified into four types depending on whether the responses or covariates are functional or scalar: (i) scalar responses with functional covariates, (ii) functional responses with scalar covariates, (iii) functional responses with functional covariates, and (iv) scalar or functional responses with functional and scalar covariates. In addition, functional regression models can be linear, partially linear, or nonlinear. In particular, functional polynomial models, functional single and multiple index models and functional additive models are three special cases of functional nonlinear models.    (1)    (2)    (3)    (4)    (5)    (6) Functional regression is a version of regression analysis when responses or covariates include functional data. Functional regression models can be classified into four types depending on whether the responses or covariates are functional or scalar: (i) scalar responses with functional covariates, (ii) functional responses with scalar covariates, (iii) functional responses with functional covariates, and (iv) scalar or functional responses with functional and scalar covariates. In addition, functional regression models can be linear, partially linear, or nonlinear. In particular, functional polynomial models, functional single and multiple index models and functional additive models are three special cases of functional nonlinear models. Functional linear models (FLMs) are an extension of linear models (LMs). A linear model with scalar response Y ∈ R {displaystyle Yin mathbb {R} } and scalar covariates X ∈ R p {displaystyle Xin mathbb {R} ^{p}} can be written as where ⟨ ⋅ , ⋅ ⟩ {displaystyle langle cdot ,cdot angle } denotes the inner product in Euclidean space, β 0 ∈ R {displaystyle eta _{0}in mathbb {R} } and β ∈ R p {displaystyle eta in mathbb {R} ^{p}} denote the regression coefficients, and ε {displaystyle varepsilon } is a random error with mean zero and finite variance. FLMs can be divided into two types based on the responses. Functional linear models with scalar responses can be obtained by replacing the scalar covariates X {displaystyle X} and the coefficient vector β {displaystyle eta } in model (1) by a centered functional covariate X c ( ⋅ ) = X ( ⋅ ) − E ( X ( ⋅ ) ) {displaystyle X^{c}(cdot )=X(cdot )-mathbb {E} (X(cdot ))} and a coefficient function β = β ( ⋅ ) {displaystyle eta =eta (cdot )} with domain T {displaystyle {mathcal {T}}} , respectively, and replacing the inner product in Euclidean space by that in Hilbert space L 2 {displaystyle L^{2}} , where ⟨ ⋅ , ⋅ ⟩ {displaystyle langle cdot ,cdot angle } here denotes the inner product in L 2 {displaystyle L^{2}} . One approach to estimating β 0 {displaystyle eta _{0}} and β ( ⋅ ) {displaystyle eta (cdot )} is to expand the centered covariate X c ( ⋅ ) {displaystyle X^{c}(cdot )} and the coefficient function β ( ⋅ ) {displaystyle eta (cdot )} in the same functional basis, for example, B-spline basis or the eigenbasis used in the Karhunen–Loève expansion. Suppose { ϕ k } k = 1 ∞ {displaystyle {phi _{k}}_{k=1}^{infty }} is an orthonormal basis of L 2 {displaystyle L^{2}} . Expanding X c {displaystyle X^{c}} and β {displaystyle eta } in this basis, X c ( ⋅ ) = ∑ k = 1 ∞ x k ϕ k ( ⋅ ) {displaystyle X^{c}(cdot )=sum _{k=1}^{infty }x_{k}phi _{k}(cdot )} , β ( ⋅ ) = ∑ k = 1 ∞ β k ϕ k ( ⋅ ) {displaystyle eta (cdot )=sum _{k=1}^{infty }eta _{k}phi _{k}(cdot )} , model (2) becomes Adding multiple functional and scalar covariates, model (2) can be extended to where Z 1 , … , Z q {displaystyle Z_{1},ldots ,Z_{q}} are scalar covariates with Z 1 = 1 {displaystyle Z_{1}=1} , α 1 , … , α q {displaystyle alpha _{1},ldots ,alpha _{q}} are regression coefficients for Z 1 , … , Z q {displaystyle Z_{1},ldots ,Z_{q}} , respectively, X j c {displaystyle X_{j}^{c}} is a centered functional covariate given by X j c ( ⋅ ) = X j ( ⋅ ) − E ( X j ( ⋅ ) ) {displaystyle X_{j}^{c}(cdot )=X_{j}(cdot )-mathbb {E} (X_{j}(cdot ))} , β j {displaystyle eta _{j}} is regression coefficient function for X j c ( ⋅ ) {displaystyle X_{j}^{c}(cdot )} , and T j {displaystyle {mathcal {T}}_{j}} is the domain of X j {displaystyle X_{j}} and β j {displaystyle eta _{j}} , for j = 1 , … , p {displaystyle j=1,ldots ,p} . However, due to the parametric component α {displaystyle alpha } , the estimation methods for model (2) cannot be used in this case and alternative estimation methods for model (3) are available. For a functional response Y ( ⋅ ) {displaystyle Y(cdot )} with domain T {displaystyle {mathcal {T}}} and a functional covariate X ( ⋅ ) {displaystyle X(cdot )} with domain S {displaystyle {mathcal {S}}} , two FLMs regressing Y ( ⋅ ) {displaystyle Y(cdot )} on X ( ⋅ ) {displaystyle X(cdot )} have been considered. One of these two models is of the form where X c ( ⋅ ) = X ( ⋅ ) − E ( X ( ⋅ ) ) {displaystyle X^{c}(cdot )=X(cdot )-mathbb {E} (X(cdot ))} is still the centered functional covariate, β 0 ( ⋅ ) {displaystyle eta _{0}(cdot )} and β ( ⋅ , ⋅ ) {displaystyle eta (cdot ,cdot )} are coefficient functions, and ε ( ⋅ ) {displaystyle varepsilon (cdot )} is usually assumed to be a random process with mean zero and finite variance. In this case, at any given time t ∈ T {displaystyle tin {mathcal {T}}} , the value of Y {displaystyle Y} , i.e., Y ( t ) {displaystyle Y(t)} , depends on the entire trajectory of X {displaystyle X} . Model (4), for any given time t {displaystyle t} , is an extension of multivariate linear regression with the inner product in Euclidean space replaced by that in L 2 {displaystyle L^{2}} . An estimating equation motivated by multivariate linear regression is