Perform an estimation of the risk on a set of curves along the sampling points.

This function performs the estimation of the risk on a set of curves along the sampling points. Both the real and estimated curves have to be sampled on the same grid.

estimate_risks(curves, curves_estim)

Arguments

curves	A list, where each element represents a real curve. Each curve have to be defined as a list with two entries: $t The sampling points $x The observed points.
curves_estim	A list, where each element represents an estimated curve. Each curve have to be defined as a list with two entries: $t The sampling points $x The estimated points.

curves

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

$t The sampling points
$x The observed points.

curves_estim

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

$t The sampling points
$x The estimated points.

Value

A list, with the mean and max integrated residual squared error in $t_0$.

Details

Actually, two risks are computed. They are defined as: $$MeanIntRSE = \frac{1}{N}\sum_{n = 1}^{N}\int(X_n(t) - \hat{X}_n(t))^2dt$$ and $$MeanIntRSE = \max_{1 \leq n \leq N} \int(X_n(t) - \hat{X}_n(t))^2dt$$

Examples

if (FALSE) {
 X <- generate_fractional_brownian(N = 1000, M = 300, H = 0.5, sigma = 0.05)
 X_smoothed <- smooth_curves(X)$smooth
 estimate_risks(X, X_smoothed)
}