Cross-validation functional regression with scalar response using kernel estimation.

Computes functional regression between functional explanatory variables and scalar response using asymmetric kernel estimation by cross-validation method.

Usage

fregre.np.cv(
  fdataobj,
  y,
  h = NULL,
  Ker = AKer.norm,
  metric = metric.lp,
  type.CV = GCV.S,
  type.S = S.NW,
  par.CV = list(trim = 0),
  par.S = list(w = 1),
  ...
)

Arguments

fdataobj: fdata class object.
y: Scalar response with length n.
h: Bandwidth, h>0. Default argument values are provided as the sequence of length 25 from 2.5%–quantile to 25%–quantile of the distance between fdataobj curves, see h.default.
Ker: Type of asymmetric kernel used, by default asymmetric normal kernel.
metric: Metric function, by default metric.lp.
type.CV: Type of cross-validation. By default generalized cross-validation GCV.S method.
type.S: Type of smothing matrix S. By default S is calculated by Nadaraya-Watson kernel estimator (S.NW).
par.CV: List of parameters for type.CV: trim, the alpha of the trimming
and draw=TRUE.
par.S: List of parameters for type.S: w, the weights.
...: Arguments to be passed for metric.lp o other metric function.

Value

Return:

call: The matched call.
residuals: y minus fitted values.
fitted.values: Estimated scalar response.
df.residual: The residual degrees of freedom.
r2: Coefficient of determination.
sr2: Residual variance.
H: Hat matrix.
y: Response.
fdataobj: Functional explanatory data.
mdist: Distance matrix between x and newx.
Ker: Asymmetric kernel used.
gcv: CV or GCV values.
h.opt: Smoothing parameter or bandwidth that minimizes CV or GCV method.
h: Vector of smoothing parameter or bandwidth.
cv: List with the fitted values and residuals estimated by CV, without the same curve.

Details

The non-parametric functional regression model can be written as follows $$ y_i =r(X_i) + \epsilon_i $$ where the unknown smooth real function $r$ is estimated using kernel estimation by means of $$\hat{r}(X)=\frac{\sum_{i=1}^{n}{K(h^{-1}d(X,X_{i}))y_{i}}}{\sum_{i=1}^{n}{K(h^{-1}d(X,X_{i}))}}$$ where $K$ is an kernel function (see Ker argument), h is the smoothing parameter and $d$ is a metric or a semi-metric (see metric argument).

The function estimates the value of smoothing parameter (also called bandwidth) h through Generalized Cross-validation GCV criteria, see GCV.S or CV.S.

The function estimates the value of smoothing parameter or the bandwidth through the cross validation methods: GCV.S or CV.S. It computes the distance between curves using the metric.lp, although any other semimetric could be used (see semimetric.basis or semimetric.NPFDA functions). Different asymmetric kernels can be used, see Kernel.asymmetric.

References

Ferraty, F. and Vieu, P. (2006). Nonparametric functional data analysis. Springer Series in Statistics, New York.

Hardle, W. Applied Nonparametric Regression. Cambridge University Press, 1994.

Febrero-Bande, M., Oviedo de la Fuente, M. (2012). Statistical Computing in Functional Data Analysis: The R Package fda.usc. Journal of Statistical Software, 51(4), 1-28. https://www.jstatsoft.org/v51/i04/

Author

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

Examples

if (FALSE) { # \dontrun{
data(tecator)
absorp=tecator$absorp.fdata
ind=1:129
x=absorp[ind,]
y=tecator$y$Fat[ind]
Ker=AKer.tri
res.np=fregre.np.cv(x,y,Ker=Ker)
summary(res.np)
res.np2=fregre.np.cv(x,y,type.CV=GCV.S,criteria="Shibata")
summary(res.np2)

## Example with other semimetrics (not run)
res.pca1=fregre.np.cv(x,y,Ker=Ker,metric=semimetric.pca,q=1)
summary(res.pca1)
res.deriv=fregre.np.cv(x,y,Ker=Ker,metric=semimetric.deriv)
summary(res.deriv)

x.d2=fdata.deriv(x,nderiv=1,method="fmm",class.out='fdata')
res.deriv2=fregre.np.cv(x.d2,y,Ker=Ker)
summary(res.deriv2)
x.d3=fdata.deriv(x,nderiv=1,method="bspline",class.out='fdata')
res.deriv3=fregre.np.cv(x.d3,y,Ker=Ker)
summary(res.deriv3)
} # }