The function uses the basis representation proposed by Ramsay and Silverman (2005) to model the relationship between the scalar response and the functional covariate by basis representation of the observed functional data \(X(t)\approx\sum_{k=1}^{k_{n1}} c_k \xi_k(t)\) and the unknown functional parameter \(\beta(t)\approx\sum_{k=1}^{k_{n2}} b_k \phi_k(t)\).
The functional linear models estimated by the expression:
$$\hat{y}= \big< X,\hat{\beta} \big> = C^{T}\psi(t)\phi^{T}(t)\hat{b}=\tilde{X}\hat{b}$$
where \(\tilde{X}(t)=C^{T}\psi(t)\phi^{T}(t)\), and \(\hat{b}=(\tilde{X}^{T}\tilde{X})^{-1}\tilde{X}^{T}y\) and so, \(\hat{y}=\tilde{X}\hat{b}=\tilde{X}(\tilde{X}^{T}\tilde{X})^{-1}\tilde{X}^{T}y=Hy\) where \(H\) is the hat matrix with degrees of freedom: \(df=tr(H)\).
If \(\lambda>0\) then fregre.basis
incorporates a roughness penalty:
\(\hat{y}=\tilde{X}\hat{b}=\tilde{X}(\tilde{X}^{T}\tilde{X}+\lambda R_0)^{-1}\tilde{X}^{T}y= H_{\lambda}y\) where \(R_0\) is the penalty matrix.
This function allows covariates of class fdata
, matrix
, data.frame
or directly covariates of class fd
. The function also gives default values to arguments basis.x
and basis.b
for representation on the basis of functional data \(X(t)\) and the functional parameter \(\beta(t)\), respectively.
If basis=
NULL
creates the bspline
basis by create.bspline.basis
.
If the functional covariate fdataobj
is a matrix or data.frame, it creates an object of class "fdata" with default attributes, see fdata
.
If basis.x$type=``fourier''
and basis.b$type=``fourier''
, the basis are orthonormal and the function decreases the number of fourier basis elements on the \(min(k_{n1},k_{n2})\), where \(k_{n1}\) and \(k_{n2}\) are the number of basis element of basis.x
and basis.b
respectively.