Perform an estimation of the risk on a set of curves at a particular point.

This function performs the estimation of the risk on a set of curves at a particular sampling points $t_0$. Both the real and estimated curves have to be sampled on the same grid.

estimate_risk(curves, curves_estim, t0_list = 0.5)

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.
t0_list	A vector of numerics, sampling points where the risk will be computed. Can have a single value.

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.

t0_list

A vector of numerics, sampling points where the risk will be computed. Can have a single value.

Value

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

Details

Actually, two risks are computed. They are defined as: $$MeanRSE(t_0) = \frac{1}{N}\sum_{n = 1}^{N}(X_n(t_0) - \hat{X}_n(t_0))^2$$ and $$MeanRSE(t_0) = \max_{1 \leq n \leq N} (X_n(t_0) - \hat{X}_n(t_0))^2$$

Examples

if (FALSE) {
 X <- generate_fractional_brownian(N = 1000, M = 300, H = 0.5, sigma = 0.05)
 X_smoothed <- smooth_curves(X)$smooth
 estimate_risk(X, X_smoothed, t0_list = 0.5)
}