Functional regression: Difference between revisions

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<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>

=== Functional additive models (FAMs) ===

Given an expansion of a functional covariate <math>X</math> onin 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 response as stated before can be written as

<math display="block">\mathbb{E}(Y|X)=\mathbb{E}(Y) + \sum_{k=1}^\infty \beta_k x_k.</math>

A functional additive model canis be givenobtained 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/>. General estimation methods have been proposed<ref>M&uuml;ller and Yao (2008). "Functional additive models". ''Journal of the American Statistical Association''. '''103''' (484):1534&ndash;1544. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1198/016214508000000751 10.1198/016214508000000751]</ref><ref>Fan, James and Radchenko (2015). "Functional additive regression". ''The Annals of Statistics''. '''43''' (5):2296&ndash;2325. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1214/15-AOS1346 10.1214/15-AOS1346]</ref>.

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/>.

== Extensions ==