fda.usc (version 2.1.0)

fregre.gsam: Fitting Functional Generalized Spectral Additive Models

Description

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

Usage

fregre.gsam(
  formula,
  family = gaussian(),
  data = list(),
  weights = NULL,
  basis.x = NULL,
  basis.b = NULL,
  ...
)

Value

Return gam object plus:

• basis.x Basis used for fdata or fd covariates.

• basis.b Basis used for beta parameter estimation.

• data List that containing the variables in the model.

• formula formula.

• y.pred predicted response by cross-validation.

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.

Author

Manuel Febrero-Bande, Manuel Oviedo de la Fuente

Details

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

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

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 \(\big<X(t),\beta\big>\) 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 gam.

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 functional beta 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.

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.

See Also

See Also as: predict.fregre.gsam and summary.gam.
Alternative methods: fregre.glm and fregre.gkam.

Examples

if (FALSE) {
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))
}