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) Functional linear models (FLMs) are an extension of linear models (LMs). A linear model with scalar response Y ∈ R {\displaystyle Y\in \mathbb {R} } and scalar covariates X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} can be written as
where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } denotes the inner product in Euclidean space, β 0 ∈ R {\displaystyle \beta _{0}\in \mathbb {R} } and β ∈ R p {\displaystyle \beta \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 Functional linear models with scalar responses can be obtained by replacing the scalar covariates X {\displaystyle X} and the coefficient vector β {\displaystyle \beta } 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 \beta =\beta (\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 \rangle } here denotes the inner product in L 2 {\displaystyle L^{2}} . One approach to estimating β 0 {\displaystyle \beta _{0}} and β ( ⋅ ) {\displaystyle \beta (\cdot )} is to expand the centered covariate X c ( ⋅ ) {\displaystyle X^{c}(\cdot )} and the coefficient function β ( ⋅ ) {\displaystyle \beta (\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 \beta } 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 \beta (\cdot )=\sum _{k=1}^{\infty }\beta _{k}\phi _{k}(\cdot )} , model (2) becomes
Y = β 0 + ∑ k = 1 ∞ β k x k + ε . {\displaystyle Y=\beta _{0}+\sum _{k=1}^{\infty }\beta _{k}x_{k}+\varepsilon .}
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