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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>Müller and Yao (2008). "Functional additive models". ''Journal of the American Statistical Association''. '''103''' (484):1534–1544. [[Digital object identifier|doi]]:[http://doi.org/10.1198/016214508000000751 10.1198/016214508000000751].</ref>. Another form of FAMs consists of a sequence of time-additive models:
<math display="block">\mathbb{E}(Y|X(t_1),\cdots,X(t_p))=\sum_{j=1}^p f_j(X(t_j)),</math>
where <math>\{t_1\cdots,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,\cdots,p</math><ref name=wang:16/><ref>Fan, James and Radchenko (2015). "Functional additive regression". ''The Annals of Statistics''. '''43''' (5):2296–2325. [[Digital object identifier|doi]]:[http://doi.org/10.1214/15-AOS1346 10.1214/15-AOS1346].</ref>.
== Extensions ==
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