Functional regression: Difference between revisions

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'''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, functionalFunctional 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 otherIn handaddition, 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 index models]] and functional [[Additive model|additive models]] are three special cases of functional nonlinear models.

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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 + \epsilon,</math>|{{EquationRef|3}}}}

where <math>Z_1,\cdots,Z_q</math> are scalar covariates with <math>Z_1=1</math>, <math>\alpha_1,\cdots,\alpha_q</math> are theregression coefficients corresponding tofor <math>Z_1,\cdots,Z_q</math>, respectively, <math>X^c_j</math> is a centered functional covariate given by <math>X_j^c(\cdot) = X_j(\cdot) - \mathbb{E}(X_j(\cdot))</math>, <math>\beta_j</math> is the correspondingregression coefficient function for <math>X_j^c(\cdot)</math>, and <math>\mathcal{T}_j</math> is the ___domain of <math>X_j</math> and <math>\beta_j</math>, for <math>j=1,\cdots,p</math>. However, due to the parametric component <math>\alpha</math>, the estimation methods for model ({{EquationNote|2}}) cannot be used in this case<ref name=wang:16>Wang, Chiou and M&uuml;ller (2016). "Functional data analysis". ''Annual Review of Statistics and Its Application''. '''3''':257&ndash;295. [[Digital object identifier|doi]]:[http://doi.org/10.1146/annurev-statistics-041715-033624 10.1146/annurev-statistics-041715-033624].</ref> and variousalternative estimation methods for model ({{EquationNote|3}}) are available<ref>Kong, Xue, Yao and Zhang (2016). "Partially functional linear regression in high dimensions". ''Biometrika''. '''103''' (1):147&ndash;159. [[Digital object identifier|doi]]:[http://doi.org/10.1093/biomet/asv062 10.1093/biomet/asv062].</ref><ref>Hu, Wang and Carroll (2004). "Profile-kernel versus backfitting in the partially linear models for longitudinal/clustered data". ''Biometrika''. '''91''' (2): 251&ndash;262. [[Digital object identifier|doi]]:[http://doi.org/10.1093/biomet/91.2.251 10.1093/biomet/91.2.251].</ref>.<br />

=== Functional linear models with functional responses ===

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A functional multiple index model is given by

<math display="block">Y = g\left(\int_{\mathcal{T}} X^c(t) \beta_1(t)dt, \cdots, \int_{\mathcal{T}} X^c(t) \beta_p(t)dt \right) + \epsilon.</math>

Taking <math>p=1</math> yields a functional single index model. However, for <math>p>1</math>, this model is problematic due to [[Curse of dimensionality|curse of dimensionality]]. With <math>p>1</math> and relatively small sample sizes, the estimator given by this model often has large variance<ref name=chen:11>Chen, Hall and M&uuml;ller (2011). "Single and multiple index functional regression models with nonparametric link". ''The Annals of Statistics''. '''39''' (3):1720&ndash;1747. [[Digital object identifier|doi]]:[http://doi.org/10.1214/11-AOS882 10.1214/11-AOS882].</ref>. An alternative <math>p</math>-component functional multiple index model can be expressed as

<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) + \epsilon.</math>

Estimation methods for functional single and multiple index models are available<ref name=chen:11/><ref>Jiang and Wang (2011). "Functional single index models for longitudinal data". '''39''' (1):362&ndash;388. [[Digital object identifier|doi]]:[http://doi.org/10.1214/10-AOS845 10.1214/10-AOS845].</ref>.