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== Functional nonlinear models ==
=== Functional polynomial models ===
Functional polynomial models are an extension of the FLMs, analogous to extending linear regression to [[Polynomial regression|polynomial regression]]. For a scalar response <math>Y</math> and a functional covariate <math>X(\cdot)</math> with ___domain <math>\mathcal{T}</math>, the simplest example of functional polynomial models is functional quadratic regression<ref name=yao:10>Yao and Müller (2010). "Functional quadratic regression". ''Biometrika''. '''97''' (1):49–64. [[Digital object identifier|doi]]:[http://doi.org/10.1093/biomet/asp069 10.1093/biomet/asp069].</ref>
<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) dsdt + \epsilon,</math>
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>\epsilon</math> is a random error with mean zero and finite variance. By analogy to FLMs, estimation of functional polynomial models can be obtained through expanding both the centered covariate <math>X^c</math> and the coefficient functions <math>\beta</math> and <math>\gamma</math> in an orthonormal basis<ref name=yao:10/>.
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