Fitting Functional Generalized Linear Models

Computes functional generalized linear model between functional covariate $X^j(t)$ (and non functional covariate $Z^j$) and scalar response $Y$ using basis representation.

Usage

fregre.glm(
  formula,
  family = gaussian(),
  data,
  basis.x = NULL,
  basis.b = NULL,
  subset = NULL,
  weights = NULL,
  ...
)

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.
family: a description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. (See family for details of family functions.)
data: List that containing the variables in the model.
basis.x: List of basis for functional explanatory data estimation.
basis.b: List of basis for $\beta(t)$ parameter estimation.
subset: an optional vector specifying a subset of observations to be used in the fitting process.
weights: weights
...: Further arguments passed to or from other methods.

Value

Return glm object plus:

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 contains the variables in the model.
formula: Formula.

Details

This function is an extension of the linear regression models: fregre.lm where the $E[Y|X,Z]$ is related to the linear prediction $\eta$ via a link function $g(.)$.

$$E[Y|X,Z]=\eta=g^{-1}(\alpha+\sum_{j=1}^{p}\beta_{j}Z^{j}+\sum_{k=1}^{q}\frac{1}{\sqrt{T_k}}\int_{T_k}{X^{k}(t)\beta_{k}(t)dt})$$

where $Z=\left[ Z^1,\cdots,Z^p \right]$ are the non functional covariates and $X(t)=\left[ X^{1}(t_1),\cdots,X^{q}(t_q) \right]$ are the functional ones.

The first item in the data list is called "df" and is a data frame with the response and non functional explanatory variables, as glm.

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 o create.basis.
basis.b is a list of basis for represent each $\beta(t)$ parameter. If basis.x is a list of functional principal components basis (see create.pc.basis or pca.fd) the argument basis.b is ignored.

represent beta lower than the number of basis used to represent the functional data.

Note

If the formula only contains a non functional explanatory variables (multivariate covariates), the function compute a standard glm procedure.

References

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

McCullagh and Nelder (1989), Generalized Linear Models 2nd ed. Chapman and Hall.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S, New York: Springer.

Author

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

Examples

if (FALSE) { # \dontrun{ 
data(tecator)
x=tecator$absorp.fdata
y=tecator$y$Fat
tt=x[["argvals"]]
dataf=as.data.frame(tecator$y)
nbasis.x=11
nbasis.b=7
basis1=create.bspline.basis(rangeval=range(tt),nbasis=nbasis.x)
basis2=create.bspline.basis(rangeval=range(tt),nbasis=nbasis.b)
f=Fat~Protein+x
basis.x=list("x"=basis1)
basis.b=list("x"=basis2)
ldata=list("df"=dataf,"x"=x)
res=fregre.glm(f,family=gaussian(),data=ldata,basis.x=basis.x,
basis.b=basis.b)
summary(res)
} # }