This section is presented as an extension of the linear regression models:
fregre.pc, fregre.pls and
fregre.basis. Now, the scalar response \(Y\) is estimated by
more than one functional covariate \(X^j(t)\) and also more than one non
functional covariate \(Z^j\). The regression model is given by:
$$E[Y|X,Z]=\alpha+\sum_{j=1}^{p}\beta_{j}Z^{j}+\sum_{k=1}^{q}\frac{1}{\sqrt{T_k}}\int_{T_k}{X^{k}(t)\beta_{k}(t)dt}
$$
where \(Z=\left[ Z^1,\cdots,Z^p \right]\) are the non
functional covariates, \(X(t)=\left[ X^{1}(t_1),\cdots,X^{q}(t_q)
\right]\) are the functional ones and
\(\epsilon\) are random errors with mean zero , finite variance
\(\sigma^2\) and \(E[X(t)\epsilon]=0\).
The first item in the data list is called "df" and is a data
frame with the response and non functional explanatory variables, as
lm. Functional covariates of class fdata or fd
are introduced in the following items in the data list.
basis.x is a list of basis for represent each functional covariate.
The basis object can be created by the function:
create.pc.basis, pca.fd
create.pc.basis, create.fdata.basis or
create.basis. basis.b is a list of basis for
represent each functional \(\beta_k\) parameter. If basis.x is a
list of functional principal components basis (see
create.pc.basis or pca.fd) the argument
basis.b (is unnecessary and) is ignored.
The user can penalty the basis elements by: (i) lambda is a list of
rough penalty values for the second derivative of each functional covariate,
see fregre.basis for more details. (ii) rn is a list
of Ridge penalty value for each functional covariate, see
fregre.pc, fregre.pls and
P.penalty for more details. Note: For the case of the
Functional Principal Components basis two penalties are allowed (but not the
two together).