Bootstrap samples of a functional statistic

provides bootstrap samples for functional data.

Usage

fdata.bootstrap(
  fdataobj,
  statistic = func.mean,
  alpha = 0.05,
  nb = 200,
  smo = 0,
  draw = FALSE,
  draw.control = NULL,
  ...
)

Arguments

fdataobj: fdata class object.
statistic: Sample statistic. It must be a function that returns an object of class fdata. By default, it uses sample mean func.mean. See Descriptive for other statistics.
alpha: Significance value.
nb: Number of bootstrap resamples.
smo: The smoothing parameter for the bootstrap samples as a proportion of the sample variance matrix.
draw: If TRUE, plot the bootstrap samples and the statistic.
draw.control: List that it specifies the col, lty and lwd for objects: fdataobj, statistic, IN and OUT.
...: Further arguments passed to or from other methods.

Value

statistic: fdata class object with the statistic estimate from nb bootstrap samples.
dband: Bootstrap estimate of (1-alpha)% distance.
rep.dist: Distance from every replicate.
resamples: fdata class object with the bootstrap resamples.
fdataobj: fdata class object.

Details

The fdata.bootstrap computes a confidence ball using bootstrap in the following way:

Let $X_1(t),\ldots,X_n(t)$ be the original data and $T=T(X_1(t),\ldots,X_n(t))$ be the sample statistic.
Calculate the nb bootstrap resamples $\left\{X_{1}^{*}(t),\cdots,X_n^*(t)\right\}$, using the following scheme: $$X_i^*(t)=X_i(t)+Z(t)$$ where $Z(t)$ is normally distributed with mean 0 and covariance matrix $\gamma\Sigma_x$, where $\Sigma_x$ is the covariance matrix of $\left\{X_1(t),\ldots,X_n(t)\right\}$ and $\gamma$ is the smoothing parameter.
Let $T^{*j}=T(X^{*j}_1(t),...,X^{*j}_n(t))$ be the estimate using the $j$ resample.
Compute $d(T,T^{*j})$, $j=1,\ldots,nb$. Define the bootstrap confidence ball of level $1-\alpha$ as $CB(\alpha)=X\in E$ such that $d(T,X)\leq d_{\alpha}$ being $d_{\alpha}$ the quantile $(1-\alpha)$ of the distances between the bootstrap resamples and the sample estimate.

The fdata.bootstrap function allows us to define a statistic calculated on the nb resamples, control the degree of smoothing by smo argument and represent the confidence ball with level $1-\alpha$ as those resamples that fulfill the condition of belonging to $CB(\alpha)$. The statistic used by default is the mean (func.mean) but also other depth-based functions can be used (see help(Descriptive)).

References

Cuevas A., Febrero-Bande, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22, 3: 481-496.

Cuevas A., Febrero-Bande, M., Fraiman R. 2006. On the use of bootstrap for estimating functions with functional data. Computational Statistics and Data Analysis 51: 1063-1074.

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
# Time consuming
#Bootstrap for Trimmed Mean with depth mode
out.boot=fdata.bootstrap(absorp,statistic=func.trim.FM,nb=200,draw=TRUE)
names(out.boot)
#Bootstrap for Median with with depth mode
control=list("col"=c("grey","blue","cyan"),"lty"=c(2,1,1),"lwd"=c(1,3,1))
out.boot=fdata.bootstrap(absorp,statistic=func.med.mode,
draw=TRUE,draw.control=control)
} # }