Smoothing of functional data using nonparametric kernel estimation

Smoothing of functional data using nonparametric kernel estimation with cross-validation (CV) or generalized cross-validation (GCV) methods.

Usage

optim.np(
  fdataobj,
  h = NULL,
  W = NULL,
  Ker = Ker.norm,
  type.CV = GCV.S,
  type.S = S.NW,
  par.CV = list(trim = 0, draw = FALSE),
  par.S = list(),
  correl = TRUE,
  verbose = FALSE,
  ...
)

Arguments

fdataobj: fdata class object.
h: Smoothing parameter or bandwidth.
W: Matrix of weights.
Ker: Type of kernel used, by default normal kernel.
type.CV: Type of cross-validation. By default generalized cross-validation (GCV) method. Possible values are GCV.S and CV.S
type.S: Type of smothing matrix S. By default S is calculated by Nadaraya-Watson kernel estimator (S.NW). Possible values are S.KNN, S.LLR, S.LPR and S.LCR.
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: tt for argvals, h for bandwidth, Ker for kernel, etc.
correl: logical. If TRUE the bandwidth parameter h is computed following the procedure described for De Brabanter et al. (2018). (option avalaible since v1.6.0 version)
verbose: If TRUE information about GCV values and input parameters is printed. Default is FALSE.
...: Further arguments passed to or from other methods. Arguments to be passed for kernel method.

Value

Returns GCV or CV values calculated for input parameters.

gcv: GCV or CV for a vector of values of the smoothing parameter h.
fdataobj: fdata class object.
fdata.est: Estimated fdata class object.
h.opt: h value that minimizes CV or GCV method.
S.opt: Smoothing matrix for the minimum CV or GCV method.
gcv.opt: Minimum of CV or GCV method.
h: Smoothing parameter or bandwidth.

Details

Calculate the minimum GCV for a vector of values of the smoothing parameter h. Nonparametric smoothing is performed by the kernel function. The type of kernel to use with the parameter Ker and the type of smothing matrix S to use with the parameter type.S can be selected by the user, see function Kernel. W is the matrix of weights of the discretization points.

Note

min.np deprecated.

References

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

Wasserman, L. All of Nonparametric Statistics. Springer Texts in Statistics, 2006.

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

De Brabanter, K., Cao, F., Gijbels, I., Opsomer, J. (2018). Local polynomial regression with correlated errors in random design and unknown correlation structure. Biometrika, 105(3), 681-69.

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{
# Exemple, phoneme DATA
data(phoneme)
mlearn<-phoneme$learn[1:100]

out1<-optim.np(mlearn,type.CV=CV.S,type.S=S.NW)
np<-ncol(mlearn)
# variance calculations
y<-mlearn
out<-out1
i<-1
z=qnorm(0.025/np)
fdata.est<-out$fdata.est
tt<-y[["argvals"]]
var.e<-Var.e(y,out$S.opt)
var.y<-Var.y(y,out$S.opt)
var.y2<-Var.y(y,out$S.opt,var.e)

# plot estimated fdata and point confidence interval
upper.var.e<-fdata.est[i,]-z*sqrt(diag(var.e))
lower.var.e<-fdata.est[i,]+z*sqrt(diag(var.e))
dev.new()
plot(y[i,],lwd=1, 
ylim=c(min(lower.var.e$data),max(upper.var.e$data)),xlab="t")
lines(fdata.est[i,],col=gray(.1),lwd=1)
lines(fdata.est[i,]+z*sqrt(diag(var.y)),col=gray(0.7),lwd=2)
lines(fdata.est[i,]-z*sqrt(diag(var.y)),col=gray(0.7),lwd=2)
lines(upper.var.e,col=gray(.3),lwd=2,lty=2)
lines(lower.var.e,col=gray(.3),lwd=2,lty=2)
legend("bottom",legend=c("Var.y","Var.error"),
col = c(gray(0.7),gray(0.3)),lty=c(1,2))
} # }