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=== Functional linear models with scalar response ===
Functional linear models with scalar response (also known as [[Generalized_functional_linear_model#Functional_linear_regression_.28FLR.29|functional linear regression (FLR)]]) can are obtained by replacing the scalar covariates <math>X</math> and the coefficient vector <math>\beta</math> in the traditional multivariate linear model by a centered functional covariate <math>X^c(t) = X(t) - \mathbb{E}(X(t))</math> and a coefficient function <math>\beta = \beta(t)</math> for <math>t\in\mathcal{T}</math>, respectively,
{{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 display="block">Y = \beta_0 + \sum_{k=1}^\infty \beta_k x_k +\epsilon,</math>
where in implementation the infinite sum is replaced by a finite sum truncated at <math>K</math>
<math display="block">Y = \beta_0 + \sum_{k=1}^K \beta_k x_k +\epsilon</math>
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