The generalized correlated cross-validation (GCCV) score.

The generalized correlated cross-validation (GCV) score.

Usage

GCCV.S(
  y,
  S,
  criteria = "GCCV1",
  W = NULL,
  trim = 0,
  draw = FALSE,
  metric = metric.lp,
  ...
)

Arguments

y: Response vectorith length n or Matrix of set cases with dimension (n x m), where n is the number of curves and m are the points observed in each curve.
S: Smoothing matrix, see S.NW, S.LLR or $S.KNN$.
criteria: The penalizing function. By default "Rice" criteria. Possible values are "GCCV1", "GCCV2", "GCCV3", "GCV".
W: Matrix of weights.
trim: The alpha of the trimming.
draw: =TRUE, draw the curves, the sample median and trimmed mean.
metric: Metric function, by default metric.lp.
...: Further arguments passed to or from other methods.

Value

Returns GCCV score calculated for input parameters.

Details

$$GCCV = \frac{\sum_{i=1}^n {(y_{i} - \hat{y}_{i, b})}^2}{(1 - \frac{tr(C)}{n})^2}$$

where $C = 2S\Sigma(\theta) - S\Sigma(\theta)S'$
and $\Sigma$ is the $n \times n$ covariance matrix with $cor(\epsilon_i, \epsilon_j) = \sigma$.

Here, $S$ is the smoothing matrix, and there are options for $C$:

A.- If $C = 2S\Sigma - S\Sigma S$
B.- If $C = S\Sigma$
C.- If $C = S\Sigma S'$

with $\Sigma$ as the $n \times n$ covariance matrix and $cor(\epsilon_i, \epsilon_j) = \sigma$.

Note

Provided that $C = I$ and the smoother matrix S is symmetric and idempotent, as is the case for many linear fitting techniques, the trace term reduces to $n - tr[S]$, which is proportional to the familiar denominator in GCV.

References

Carmack, P. S., Spence, J. S., and Schucany, W. R. (2012). Generalised correlated cross-validation. Journal of Nonparametric Statistics, 24(2):269–282.

Oviedo de la Fuente, M., Febrero-Bande, M., Munoz, P., and Dominguez, A. Predicting seasonal influenza transmission using Functional Regression Models with Temporal Dependence.https://arxiv.org/abs/1610.08718

Author

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

Examples

if (FALSE) { # \dontrun{
data(tecator)
x=tecator$absorp.fdata
x.d2<-fdata.deriv(x,nderiv=)
tt<-x[["argvals"]]
dataf=as.data.frame(tecator$y)
y=tecator$y$Fat
# plot the response
plot(ts(tecator$y$Fat))

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)
basis.x=list("x.d2"=basis1)
basis.b=list("x.d2"=basis2)
ldata=list("df"=dataf,"x.d2"=x.d2)
# No correlation
res.gls=fregre.gls(Fat~x.d2,data=ldata, 
                   basis.x=basis.x,basis.b=basis.b)
# AR1 correlation                   
res.gls=fregre.gls(Fat~x.d2,data=ldata, correlation=corAR1(),
                   basis.x=basis.x,basis.b=basis.b)
GCCV.S(y,res.gls$H,"GCCV1",W=res.gls$W)
res.gls$gcv
} # }