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{{Short description|Type of regression analysis}}
'''Functional regression''' is a version of [[
▲'''Functional regression''' is a version of [[Regression analysis|regression analysis]] when [[Dependent and independent variables|responses]] or [[Dependent and independent variables|covariates]] include [[Functional data analysis|functional data]]. On the one hand, 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. On the other hand, functional regression models can be [[Linear regression|linear]], partially linear, or [[Nonlinear regression|nonlinear]]. In particular, functional polynomial models, functional [[Semiparametric_regression#Index_models|single and multiple single models]] and functional [[Additive model|additive models]] are three special cases of functional nonlinear models.
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== Functional linear models (FLMs) ==
Functional linear models (FLMs) are an extension of [[Linear regression|linear models]] (LMs). A linear model with scalar response <math>Y\in\mathbb{R}</math> and scalar covariates <math>X\in\mathbb{R}^p</math> can be written as
{{NumBlk|::|<math display="block">Y = \beta_0 + \langle X,\beta\rangle + \
where <math>\langle\cdot,\cdot\rangle</math> denotes the [[Inner product space|inner product]] in [[
=== Functional linear models with scalar responses ===
Functional linear models with scalar responses can be obtained by replacing the scalar covariates <math>X</math> and the coefficient vector <math>\beta</math> in model ({{EquationNote|1}}) by a centered functional covariate <math>X^c(\cdot) = X(\cdot) - \mathbb{E}(X(\cdot))</math> and a coefficient function <math>\beta = \beta(\cdot)</math> with [[Domain of a function|___domain]] <math>\mathcal{T}</math>, respectively, and replacing the inner product in Euclidean space by that in [[Hilbert space]] [[Lp space|<math>L^2</math>]],
{{NumBlk|::|<math display="block">Y = \beta_0 + \langle X^c, \beta\rangle +\
where <math>\langle \cdot, \cdot \rangle</math> here denotes the inner product in <math>L^2</math>. One approach to estimating <math>\beta_0</math> and <math>\beta(\cdot)</math> is to expand the centered covariate <math>X^c(\cdot)</math> and the coefficient function <math>\beta(\cdot)</math> in the same [[Basis function|functional basis]], for example, [[
<math display="block">Y = \beta_0 + \sum_{k=1}^\infty \beta_k x_k +\
For implementation, regularization is needed and can be done through truncation, <math>L^2</math> penalization or <math>L^1</math> penalization.<ref name=morr:15>
Adding multiple functional and scalar covariates, model ({{EquationNote|2}}) can be extended to
{{NumBlk|::|<math display="block">Y = \sum_{k=1}^q Z_k\alpha_k + \sum_{j=1}^p \int_{\mathcal{T}_j} X_j^c(t) \beta_j(t) \,dt + \
where <math>Z_1,\
=== Functional linear models with functional responses ===
For a functional response <math>Y(\cdot)</math> with ___domain <math>\mathcal{T}</math> and a functional covariate <math>X(\cdot)</math> with ___domain <math>\mathcal{S}</math>, two FLMs regressing <math>Y(\cdot)</math> on <math>X(\cdot)</math> have been considered.<ref name=wang:16/><ref>Ramsay and [[Bernard Silverman|Silverman]] (2005). ''Functional data analysis'', 2nd ed., New York
{{NumBlk|::|<math display="block">Y(t) = \beta_0(t) + \int_{\mathcal{S}} \beta(s,t) X^c(s)\,ds + \
where <math>X^c(\cdot) = X(\cdot) - \mathbb{E}(X(\cdot))</math> is still the centered functional covariate, <math>\beta_0(\cdot)</math> and <math>\beta(\cdot,\cdot)</math> are coefficient functions, and <math>\
<math display="block">r_{XY} = R_{XX}\beta, \text{ for } \beta\in L^2(\mathcal{S}\times\mathcal{S}),</math>
where <math>r_{XY}(s,t) = \text{cov}(X(s),Y(t))</math>, <math>R_{XX}: L^2(\mathcal{S}\times\mathcal{S}) \rightarrow L^2(\mathcal{S}\times\mathcal{T})</math> is defined as <math>(R_{XX}\beta)(s,t) = \int_\mathcal{S} r_{XX}(s,w)\beta(w,t)dw</math> with <math>r_{XX}(s,w) = \text{cov}(X(s),X(w))</math> for <math>s,w\in\mathcal{S}</math>.<ref name=wang:16/>
When <math>X</math> and <math>Y</math> are concurrently observed, i.e., <math>\mathcal{S}=\mathcal{T}</math>,<ref>{{Cite journal |last=Grenander
Adding multiple functional covariates, model ({{EquationNote|4}}) can be extended to
{{NumBlk|::|<math display="block">Y(t) = \beta_0(t) + \sum_{j=1}^p\int_{\mathcal{S}_j} \beta_j(s,t) X^c_j(s)\,ds + \
where for <math>j=1,\
<math display="block">Y(t) = \sum_{j=1}^p X_j \beta_j(t) + \
which is a FLM with functional responses and scalar covariates.
==== Functional concurrent models ====
Assuming that <math>\mathcal{S} = \mathcal{T}</math>, another model, known as the functional concurrent model, sometimes also referred to as the varying-coefficient model, is of the form
{{NumBlk|::|<math display="block">Y(t) = \alpha_0(t) + \alpha(t)X(t)+\
where <math>\alpha_0</math> and <math>\alpha</math> are coefficient functions. Note that model ({{EquationNote|6}}) assumes the value of <math>Y</math> at time <math>t</math>, i.e., <math>Y(t)</math>, only depends on that of <math>X</math> at the same time, i.e., <math>X(t)</math>. Various estimation methods can be applied to model ({{EquationNote|6}}).<ref>{{Cite journal |last=Fan
Adding multiple functional covariates, model ({{EquationNote|6}}) can also be extended to
<math display="block">Y(t) = \alpha_0(t) + \sum_{j=1}^p\alpha_j(t)X_j(t)+\
where <math>X_1,\
== Functional nonlinear models ==
=== Functional polynomial models ===
Functional polynomial models are an extension of the FLMs with scalar responses, analogous to extending linear regression to [[
<math display="block">Y = \alpha + \int_\mathcal{T}\beta(t)X^c(t)\,dt + \int_\mathcal{T} \int_\mathcal{T} \gamma(s,t) X^c(s)X^c(t)
where <math>X^c(\cdot) = X(\cdot) - \mathbb{E}(X(\cdot))</math> is the centered functional covariate, <math>\alpha</math> is a scalar coefficient, <math>\beta(\cdot)</math> and <math>\gamma(\cdot,\cdot)</math> are coefficient functions with domains <math>\mathcal{T}</math> and <math>\mathcal{T}\times\mathcal{T}</math>, respectively, and <math>\
=== Functional single and multiple index models ===
A functional multiple index model is given by
<math display="block">Y = g\left(\int_{\mathcal{T}} X^c(t) \beta_1(t)\,dt, \
Taking <math>p=1</math> yields a functional single index model. However, for <math>p>1</math>, this model is problematic due to [[
<math display="block">Y = g_1\left(\int_{\mathcal{T}} X^c(t) \beta_1(t)\,dt\right)+ \cdots+ g_p\left(\int_{\mathcal{T}} X^c(t) \beta_p(t)\,dt \right) + \
Estimation methods for functional single and multiple index models are available.<ref name=chen:11/><ref>{{Cite journal |last=Jiang
=== Functional additive models (FAMs) ===
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One form of FAMs is obtained by replacing the linear function of <math>x_k</math>, i.e., <math>\beta_k x_k</math>, by a general smooth function <math>f_k</math>,
<math display="block">\mathbb{E}(Y|X)=\mathbb{E}(Y) + \sum_{k=1}^\infty f_k(x_k),</math>
where <math>f_k</math> satisfies <math>\mathbb{E}(f_k(x_k))=0</math> for <math>k\in\mathbb{N}</math>.<ref name=wang:16/><ref>
<math display="block">\mathbb{E}(Y|X(t_1),\
where <math>\{t_1,\
== Extensions ==
A direct extension of FLMs with scalar responses shown in model ({{EquationNote|2}}) is to add a link function to create a [[
# Linear predictor <math>\eta = \beta_0 + \int_{\mathcal{T}} X^c(t)\beta(t)\,dt</math>;
# [[Variance function]] <math>\text{Var}(Y|X) = V(\mu)</math>, where <math>\mu = \mathbb{E}(Y|X)</math> is the [[Conditional expectation|conditional mean]];
# Link function <math>g</math> connecting the conditional mean and the linear predictor through <math>\mu=g(\eta)</math>.
== See also ==
* [[Functional data analysis
* [[Functional * [[Karhunen–Loève theorem]]
▲* [[Generalized functional linear model|Generalized functional linear model]]
▲* [[Stochastic processes|Stochastic processes]]
== References ==
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