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Functional regression with scalar response using non-parametric kernel estimation

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

Computes functional regression between functional explanatory variables and scalar response using kernel estimation.

Usage

fregre.np(
  fdataobj,
  y,
  h = NULL,
  Ker = AKer.norm,
  metric = metric.lp,
  type.S = S.NW,
  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 5%–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.S

Type of smothing matrix S. By default S is calculated by Nadaraya-Watson kernel estimator (S.NW).

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.

  • fitted.values: Estimated scalar response.

  • H: Hat matrix.

  • residuals: y minus fitted values.

  • df.residual: The residual degrees of freedom.

  • r2: Coefficient of determination.

  • sr2: Residual variance.

  • y: Response.

  • fdataobj: Functional explanatory data.

  • mdist: Distance matrix between x and newx.

  • Ker: Asymmetric kernel used.

  • h.opt: Smoothing parameter or bandwidth.

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 distance between curves is calculated using the metric.lp although any other semimetric could be used (see semimetric.basis or semimetric.NPFDA functions). The kernel is applied to a metric or semi-metrics that provides non-negative values, so it is common to use asymmetric kernels. 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.

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/

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

See also

See Also as: fregre.np.cv, summary.fregre.fd and predict.fregre.fd .
Alternative method: fregre.basis,cand fregre.pc.

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]

res.np=fregre.np(x,y,Ker=AKer.epa)
summary(res.np)
res.np2=fregre.np(x,y,Ker=AKer.tri)
summary(res.np2)

# with other semimetrics.
res.pca1=fregre.np(x,y,Ker=AKer.tri,metri=semimetric.pca,q=1)
summary(res.pca1)
res.deriv=fregre.np(x,y,metri=semimetric.deriv)
summary(res.deriv)
x.d2=fdata.deriv(x,nderiv=1,method="fmm",class.out='fdata')
res.deriv2=fregre.np(x.d2,y)
summary(res.deriv2)
x.d3=fdata.deriv(x,nderiv=1,method="bspline",class.out='fdata')
res.deriv3=fregre.np(x.d3,y)
summary(res.deriv3)
} # }