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{{Short description|Type of regression analysis}}
'''Functional regression''' is a version of [[
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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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