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Fitting Functional Generalized Kernel Additive Models.

Source: R/fregre.gkam.r
fregre.gkam.Rd

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"),
  ...
)

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 procedure only considers functional covariates (not implemented for non-functional covariates). 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

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

...

Further arguments passed to or from other methods.

inverse

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

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. doi: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

Author

Febrero-Bande, M. and Oviedo de la Fuente, M.

Examples

if (FALSE) { # \dontrun{
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
} # }