fda.usc (version 1.1.0)

fregre.gkam: Fitting Functional Generalized Kernel Additive Models.

Description

Computes functional regression between functional explanatory variables $(X^{1}(t_1),...,X^{q}(t_q))$ and scalar response $Y$ using backfitting algorithm.

Usage

fregre.gkam(formula,family = gaussian(),data, weights= rep(1,nobs),
par.metric = NULL,par.np=NULL,offset=NULL,
control = list(maxit = 100,epsilon = 0.001, trace = FALSE, inverse="solve"),...)
kgam.H(object,inverse="svd")

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.

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

weights

weights

par.metric

List of arguments by covariate to pass to the metric function by covariate.

par.np

List of arguments to pass to the fregre.np.cv function

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting.

control

a list of parameters for controlling the fitting process, by default: maxit, epsilon, trace and inverse

object

List that containing the Hat matrix for each variable in the model.

inverse

="svd" (by default) or ="solve" method.

...

Further arguments passed to or from other methods.

Value

• result
  List of non-parametric estimation by covariate.
• fitted.values
  Estimated scalar response.
• residuals
  y minus fitted values.
• effects
  The residual degrees of freedom.
• alpha
  Hat matrix.
• family
  Coefficient of determination.
• linear.predictors
  Residual variance.
• deviance
  Scalar response.
• aic
  Functional explanatory data.
• null.deviance
  Non functional explanatory data.
• iter
  Distance matrix between curves.
• w
  beta coefficient estimated
• eqrank
  List that containing the variables in the model.
• prior.weights
  Asymmetric kernel used.
• y
  Scalar response.
• H
  Hat matrix, see Opsomer and Ruppert(1997) for more details.
• converged
  conv.

Details

The smooth functions $f(.)$ are estimated nonparametrically using a iterative local scoring algorithm by applying Nadaraya-Watson weighted kernel smoothers using fregre.np.cv in each step, see Febrero-Bande and Gonzalez-Manteiga (2011) for more details. Consider the fitted response $\hat{Y}=g^{-1}(H_{Q}y)$, where $H_{Q}$ is the weighted hat matrix. Opsomer and Ruppert (1997) solves a system of equations for fit the unknowns $f(\cdot)$ computing the additive smoother matrix $H_k$ such that $\hat{f}_k (X^k)=H_{k}Y$ and $H_Q=H_1+,\cdots,+H_q$. The additive model is fitted as follows: $$\hat{Y}=g^{-1}\Big(\sum_i^q \hat{f_i}(X_i)\Big)$$

References

Febrero-Bande M. and Gonzalez-Manteiga W. (2012). Generalized Additive Models for Functional Data. TEST. Springer-Velag. http://dx.doi.org/10.1007/s11749-012-0308-0 Opsomer J.D. and Ruppert D.(1997). Fitting a bivariate additive model by local polynomial regression.Annals of Statistics, 25, 186-211.

See Also

See Also as: fregre.gsam, fregre.glm and fregre.np.cv

Examples

data(tecator)
 ab=tecator$absorp.fdata[1:100]
 ab2=fdata.deriv(ab,2)
 yfat=tecator$y[1:100,"Fat"]

## Example 1: # Changing the argument par.np and family
 yfat.cat=ifelse(yfat<15,0,1)
 xlist=list("df"=data.frame(yfat.cat),"ab"=ab,"ab2"=ab2)
 f2<-yfat.cat~ab+ab2

 par.NP<-list("ab"=list(Ker=AKer.norm,type.S="S.NW"),
 "ab2"=list(Ker=AKer.norm,type.S="S.NW"))
# res2=fregre.gkam(f2,family=binomial(),data=xlist,
# par.np=par.NP)
# res2

## Example 2: Changing the argument par.metric and family link
 par.metric=list("ab"=list(metric=semimetric.deriv,nderiv=2,nbasis=15),
 "ab2"=list("metric"=semimetric.basis))
# res3=fregre.gkam(f2,family=binomial("probit"),data=xlist,
# par.metric=par.metric,control=list(maxit=2,trace=FALSE))
# summary(res3)

## Example 3: Gaussian family (by default)
## Only 1 iteration (by default maxit=100)
# xlist=list("df"=data.frame(yfat),"ab"=ab,"ab2"=ab2)
# f<-yfat~ab+ab2
# res=fregre.gkam(f,data=xlist,control=list(maxit=1,trace=FALSE))
# res