fregre.lm: Fitting Functional Linear Models

Description

Computes functional regression between functional (and non functional) explanatory variables and scalar response using basis representation.

Usage

fregre.lm(
  formula,
  data,
  basis.x = NULL,
  basis.b = NULL,
  lambda = NULL,
  P = NULL,
  weights = rep(1, n),
  ...
)

Value

Return lm object plus:

sr2 Residual variance.
Vp Estimated covariance matrix for the parameters.
lambda A roughness penalty.
basis.x Basis used for fdata or fd covariates.
basis.b Basis used for beta parameter estimation.
beta.l List of estimated beta parameter of functional covariates.
data List that containing the variables in the model.
formula formula.

Arguments

formula: an object of class formula (or one that can be coerced to that class): a symbolic description of the model to be fitted. The details of model specification are given under Details.
data: List that containing the variables in the model. Functional covariates are recommended to be of class fdata. Objects of class "fd" can be used at the user's own risk.
basis.x: List of basis for functional explanatory data estimation.
basis.b: List of basis for functional beta parameter estimation.
lambda: List, indexed by the names of the functional covariates, which contains the Roughness penalty parameter.
P: List, indexed by the names of the functional covariates, which contains the parameters for the creation of the penalty matrix.
weights: weights
...: Further arguments passed to or from other methods.

Author

Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@usc.es

Details

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.

Penalty options are under development, not guaranteed to work properly. The user can penalty the basis elements by: (i) lambda is a list of rough penalty values of each functional covariate, see P.penalty for more details.

References

Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.

Febrero-Bande, M., Oviedo de la Fuente, M. (2012). Statistical Computing in Functional Data Analysis: The R Package fda.usc. Journal of Statistical Software, 51(4), 1-28. https://www.jstatsoft.org/v51/i04/