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{{NumBlk|::|<math display="block">Y = \beta_0 + \langle X^c, \beta\rangle +\epsilon = \beta_0 + \int_\mathcal{T} X^c(t)\beta(t)dt + \epsilon,</math>|{{EquationRef|1}}}}
where <math>\langle \cdot, \cdot \rangle</math> here denotes the inner product in [[Lp space|<math>L^2</math> space]]. One approach to estimating <math>\beta_0</math> and <math>\beta(t)</math> is to expand the covariate <math>X</math> and the coefficient function <math>\beta(t)</math> in the same [[Basis function|functional basis]], such as [[B-spline|B-spline]] basis or the eigenfunctions in the [[Karhunen–Loève theorem|Karhunen–Loève expansion]]. Suppose <math>\{\phi_k\}_{k=1}^\infty</math> is an [[Orthonormal basis|orthonormal basis]] of <math>L^2</math> space. Expanding <math>X</math> and <math>\beta</math> in this basis, <math>X^c(t) = \sum_{k=1}^\infty x_k \phi_k(t)</math>, <math>\beta(t) = \sum_{k=1}^\infty \beta_k \phi_k(t)</math>, model ({{EquationNote|1}}) becomes
{{NumBlk|::|<math display="block">Y = \beta_0 + \sum_{k=1}^\infty \beta_k x_k +\epsilon
{{NumBlk|::|<math display="block">Y = \beta_0 + \sum_{k=1}^K \beta_k x_k +\epsilon,</math>|{{EquationRef|3}}}}
where <math>K\in\mathbb{N}</math> is finite<ref name=wang:16>Wang, Chiou and Müller (2016). "Functional data analysis". ''Annual Review of Statistics and Its Application''. '''3''':257–295. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1146/annurev-statistics-041715-033624 10.1146/annurev-statistics-041715-033624]</ref>. Model ({{EquationNote|3}}) is nothing but linear regression with response <math>Y</math> and covariates <math>x_1,\cdots,x_K</math>. Estimation of <math>\beta_k,\ k=0,1,\cdots,K</math> can be obtained by [[Least squares|ordinary least squares]].<br />
Adding multiple functional and scalar covariates, the FLR can be extended as
<math display="block">Y = \langle\mathbf{Z},\alpha\rangle + \sum_{j=1}^p \int_{\mathcal{T}_j} X_j^c(t) \beta_j(t) dt + \epsilon</math>
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