Perform a pre-smoothing of the data.

Source: R/estimate_parameters.R

This function performs a pre-smoothing of the data using a Nadaraya-Watson estimator. We use an Epanechnikov kernel and a naive bandwidth.

presmoothing(
  curves,
  point = 0.5,
  delta_f = NULL,
  kernel = "epanechnikov",
  beta = 1,
  bandwidth_naive = 0
)

Arguments

curves

List, where each element represents a curve. Each curve have to be defined as a list with two entries:

  • $t Sampling points

  • $x Observed points

point

Numeric (default = 0.5), sampling point at which the data is pre-smoothed.

delta_f

Function (default = NULL), function to determine the delta.

kernel

String (default = 'epanechnikov'), the kernel used for the estimation:

  • epanechnikov

  • uniform

  • biweight

beta

Numeric (default = 1), pre-specified regularity of the curves. The default value is 1, which correspond to at least one time differentiable curves.

bandwidth_naive

Numeric (default = 0), bandwidth to use for the presmoothing. If set to 0, the bandwidth will be defined as $$\frac{\delta}{m}^{1 / (2\beta + 1)}$$ where

  • \(m\) is the mean number of sampling points per curve.

  • \(\delta\) is the length of the interval where the smoothing is done.

  • \(\beta\) represents the regularity of the curves.

Value

List, with two entries:

  • $grid Grid on which the smoothing has been done.

  • $x_smooth The smoothed data.

References

S. Golovkine, N. Klutchnikoff and V. Patilea (2021) - Adaptive optimal estimation of irregular mean and covariance functions.