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{{Short description|Type of regression analysis}}
'''Functional regression''' is a version of
▲'''Functional regression''' is a version of the [[Regression analysis|regression analysis]] when [[Dependent and independent variables|responses]] or [[Dependent and independent variables|covariates]] include [[Functional data analysis|functional data]]. One the one hand, functional regression models can be classified into four types depending on whether the response or covariates are functional or scalar: (i) scalar response with functional covariates, (ii) functional response with scalar covariates, (iii) functional response with functional covariates, and (iv) scalar or functional response 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
{{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
Functional linear models with scalar
{{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
Adding multiple functional and scalar covariates,
{{NumBlk|::|<math display="block">Y = \
where <math>
=== Functional linear models with functional
For a functional response <math>Y(\cdot)</math>
{{NumBlk|::|<math display="block">Y(
where
<math display="block">r_{XY} = R_{XX}\beta, \text{ for } \beta\in L^2(\mathcal{
where <math>r_{XY}(s,t) = \text{cov}(X(s),Y(t))</math>, <math>R_{XX}: L^2(\mathcal{S}\times
When <math>X</math> and <math>Y</math> are concurrently observed, i.e., <math>\mathcal{S}=\mathcal{T}</math>,<ref>{{Cite journal |last=Grenander |first=Ulf |date= |title=Stochastic processes and statistical inference |url=https://projecteuclid.org/journals/arkiv-for-matematik/volume-1/issue-3/Stochastic-processes-and-statistical-inference/10.1007/BF02590638.full |journal=Arkiv för Matematik |volume=1 |issue=3 |pages=195–277 |doi=10.1007/BF02590638 |issn=0004-2080}}</ref> it is reasonable to consider a historical functional linear model, where the current value of <math>Y</math> only depends on the history of <math>X</math>, i.e., <math>\beta(s,t)=0</math> for <math>s>t</math> in model ({{EquationNote|4}}).<ref name=wang:16/><ref>{{Cite journal |last=Malfait |first=Nicole |last2=Ramsay |first2=James O. |date=2003 |title=The historical functional linear model |url=https://onlinelibrary.wiley.com/doi/10.2307/3316063 |journal=Canadian Journal of Statistics |language=en |volume=31 |issue=2 |pages=115–128 |doi=10.2307/3316063 |issn=1708-945X|url-access=subscription }}</ref> A simpler version of the historical functional linear model is the functional concurrent model (see below).<br />
Adding multiple functional covariates, model ({{EquationNote|4}}) can be extended to
<math display="block">Y(s) = \sum_{j=1}^p X_j \beta_j(s) + \epsilon(s)</math>▼
{{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 + \varepsilon(t),\ \text{for}\ t\in\mathcal{T},</math>|{{EquationRef|5}}}}
which is a FLM with functional response and scalar covariates.▼
where for <math>j=1,\ldots,p</math>, <math>X_j^c(\cdot)=X_j(\cdot) - \mathbb{E}(X_j(\cdot))</math> is a centered functional covariate with ___domain <math>\mathcal{S}_j</math>, and <math>\beta_j(\cdot,\cdot)</math> is the corresponding coefficient function with the same ___domain, respectively.<ref name=wang:16/> In particular, taking <math>X_j(\cdot)</math> as a constant function yields a special case of model ({{EquationNote|5}})
▲<math display="block">Y(
====
Assuming that <math>\mathcal{
{{NumBlk|::|<math display="block">Y(
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 |first=Jianqing |last2=Zhang |first2=Wenyang |date= |title=Statistical estimation in varying coefficient models |url=https://projecteuclid.org/journals/annals-of-statistics/volume-27/issue-5/Statistical-estimation-in-varying-coefficient-models/10.1214/aos/1017939139.full |journal=The Annals of Statistics |volume=27 |issue=5 |pages=1491–1518 |doi=10.1214/aos/1017939139 |issn=0090-5364}}</ref><ref>{{Cite journal |last=Huang |first=Jianhua Z. |last2=Wu |first2=Colin O. |last3=Zhou |first3=Lan |date=2004 |title=Polynomial Spline Estimation and Inference for Varying Coefficient Models with Longitudinal Data |url=https://www.jstor.org/stable/24307415 |journal=Statistica Sinica |volume=14 |issue=3 |pages=763–788 |issn=1017-0405}}</ref><ref>{{Cite journal |last=Şentürk |first=Damla |last2=Müller |first2=Hans-Georg |date=2010-09-01 |title=Functional Varying Coefficient Models for Longitudinal Data |url=https://www.tandfonline.com/doi/abs/10.1198/jasa.2010.tm09228 |journal=Journal of the American Statistical Association |doi=10.1198/jasa.2010.tm09228 |issn=0162-1459|url-access=subscription }}</ref><br />
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)+\varepsilon(t),\ \text{for}\ t\in\mathcal{T},</math>
where <math>X_1,\ldots,X_p</math> are multiple functional covariates with ___domain <math>\mathcal{T}</math> and <math>\alpha_0,\alpha_1,\ldots,\alpha_p</math> are the coefficient functions with the same ___domain.<ref name=wang:16/>
== Functional nonlinear models ==
=== Functional polynomial models ===
Functional polynomial models are an extension of the FLMs with scalar responses, analogous to extending linear regression to [[polynomial regression]]. For a scalar response <math>Y</math> and a functional covariate <math>X(\cdot)</math>
<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
=== 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 |first=Ci-Ren |last2=Wang |first2=Jane-Ling |date= |title=Functional single index models for longitudinal data |url=https://projecteuclid.org/journals/annals-of-statistics/volume-39/issue-1/Functional-single-index-models-for-longitudinal-data/10.1214/10-AOS845.full |journal=The Annals of Statistics |volume=39 |issue=1 |pages=362–388 |doi=10.1214/10-AOS845 |issn=0090-5364|arxiv=1103.1726 }}</ref>
=== Functional additive models (FAMs) ===
Given an expansion of a functional covariate <math>X</math> with ___domain <math>\mathcal{T}</math> in an orthonormal basis <math>\{\phi_k\}_{k=1}^\infty</math>: <math>X(t) = \sum_{k=1}^\infty x_k \phi_k(t)</math>, a functional linear model with scalar
<math display="block">\mathbb{E}(Y|X)=\mathbb{E}(Y) + \sum_{k=1}^\infty \beta_k x_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/>
<math display="block">\mathbb{E}(Y|X(t_1),\ldots,X(t_p))=\sum_{j=1}^p f_j(X(t_j)),</math>
where <math>\{t_1,\ldots,t_p\}</math> is a dense grid on <math>\mathcal{T}</math> with increasing size <math>p\in\mathbb{N}</math>, and <math>f_j(x) = g(t_j,x)</math> with <math>g</math> a smooth function, for <math>j=1,\ldots,p</math><ref name=wang:16/><ref>{{Cite journal |last=Fan |first=Yingying |last2=James |first2=Gareth M. |last3=Radchenko |first3=Peter |date= |title=Functional additive regression |url=https://projecteuclid.org/journals/annals-of-statistics/volume-43/issue-5/Functional-additive-regression/10.1214/15-AOS1346.full |journal=The Annals of Statistics |volume=43 |issue=5 |pages=2296–2325 |doi=10.1214/15-AOS1346 |issn=0090-5364|arxiv=1510.04064 }}</ref>
== Extensions ==
A direct extension of
# Linear predictor <math>\eta
# [[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 linear model|Generalized linear model]]
▲* [[Stochastic processes|Stochastic processes]]
== References ==
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