Fitting Functional Generalized Spectral Additive Models

Computes a functional GAM model between a functional covariate $(X^1(t_1), \dots, X^{q}(t_q))$ and a non-functional covariate $(Z^1, ..., Z^p)$ with a scalar response $Y$.

This function extends functional generalized linear regression models (fregre.glm) where $E[Y|X,Z]$ is related to the linear predictor $\eta$ via a link function $g(\cdot)$ with integrated smoothness estimation by the smooth functions $f(\cdot)$.

$$E[Y|X,Z]=\eta=g^{-1}\left(\alpha+\sum_{i=1}^{p}f_{i}(Z^{i})+\sum_{k=1}^{q}\sum_{j=1}^{k_q} f_{j}^{k}(\xi_j^k)\right)$$

where $\xi_j^k$ is the coefficient of the basis function expansion of $X^k$; in PCA analysis, $\xi_j^k$ is the score of the $j$-functional PC of $X^k$.

Usage

fregre.gsam(
  formula,
  family = gaussian(),
  data = list(),
  weights = NULL,
  basis.x = NULL,
  basis.b = 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.
weights: weights
basis.x: List of basis for functional explanatory data estimation.
basis.b: List of basis for functional beta parameter estimation.
...: Further arguments passed to or from other methods.

Value

Return gam object plus:

basis.x: Basis used for fdata or fd covariates.
basis.b: Basis used for beta parameter estimation.
data: List containing the variables in the model.
formula: Formula used in the model.
y.pred: Predicted response by cross-validation.

Details

The smooth functions $f(\cdot)$ can be added to the right-hand side of the formula to specify that the linear predictor depends on smooth functions of predictors using smooth terms s and te as in gam (or linear functionals of these as $Z \beta$ and $\langle X(t), \beta \rangle$ in fregre.glm).

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

Functional covariates of class fdata or fd are introduced in the following items of the data list.
basis.x is a list of basis functions for representing each functional covariate. The basis object can be created using functions such as create.pc.basis, pca.fd, create.fdata.basis, or create.basis.
basis.b is a list of basis functions for representing each functional beta parameter. If basis.x is a list of functional principal components basis functions (see create.pc.basis or pca.fd), the argument basis.b is ignored.

Note

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

References

Muller HG and Stadtmuller U. (2005). Generalized functional linear models. Ann. Statist.33 774-805.

Wood (2001) mgcv:GAMs and Generalized Ridge Regression for R. R News 1(2):20-25.

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

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$absorp.fdata
x.d1 <- fdata.deriv(x)
tt <- x[["argvals"]]
dataf <- as.data.frame(tecator$y)
nbasis.x <- 11
nbasis.b <- 5
basis1 <- create.bspline.basis(rangeval=range(tt),nbasis=nbasis.x)
basis2 <- create.bspline.basis(rangeval=range(tt),nbasis=nbasis.b)
f <- Fat ~ s(Protein) + s(x)
basis.x <- list("x"=basis1,"x.d1"=basis1)
basis.b <- list("x"=basis2,"x.d1"=basis2)
ldat <- ldata("df" = dataf, "x" = x , "x.d1" = x.d1)
res <- fregre.gsam(Fat ~ Water + s(Protein) + x + s(x.d1), ldat,
                   family = gaussian(),  basis.x = basis.x,
                   basis.b = basis.b)
summary(res)
pred <- predict(res,ldat)
plot(pred-res$fitted)
pred2 <- predict.gam(res,res$XX)
plot(pred2-res$fitted)
plot(pred2-pred)
res2 <- fregre.gsam(Fat ~ te(Protein, k = 3) + x, data =  ldat,
                     family=gaussian())
summary(res2)

##  dropind basis pc
basis.pc0 <- create.pc.basis(x,c(2,4,7))
basis.pc1 <- create.pc.basis(x.d1,c(1:3))
basis.x <- list("x"=basis.pc0,"x.d1"=basis.pc1)
ldata <- ldata("df"=dataf,"x"=x,"x.d1"=x.d1)  
res.pc <- fregre.gsam(f,data=ldata,family=gaussian(),
          basis.x=basis.x,basis.b=basis.b)
summary(res.pc)
 
##  Binomial family
ldat$df$FatCat <- factor(ifelse(tecator$y$Fat > 20, 1, 0))
res.bin <- fregre.gsam(FatCat ~ Protein + s(x),ldat,family=binomial())
summary(res.bin)
table(ldat$df$FatCat, ifelse(res.bin$fitted.values < 0.5,0,1))
} # }