conf_band {edecob}

Confidence Bounds of the Smoother

Description

Calculate the confidence bounds of the smoother function using the bootstrap.

Usage

conf_band(bt_smoother, smoother_pts, bt_tot_rep, conf_band_lvl)

Arguments

bt_smoother

A data frame containing the bootstrapped smoother. Use the output of bt_smoother.

smoother_pts

A data frame containing the smoother. Preferably the output of one of the smoother functions included in this package.

bt_tot_rep

The number of iterations for the bootstrap computation. Because of run time, it is recommended to keep this number below 500. Defaults to 100.

conf_band_lvl

The confidence level for the simultaneous confidence bands. Defaults to 0.95. When detection of events using only the smoother is desired, conf_band_lvl can be chosen to be 0.

Details

The procedure is as follows:

1. We compute the quantiles

   q_x(t_i), q_{1-x}(t_i) i = 1,\dots, N

   where

   q_x(t_i) = inf\left\{u; P^*[S(t_i)^*_b - S(t_i) \le u] \ge x\right\}

   is a pointwise bootstrap quantile, S(t_i)^*_b the bootstrapped smoother, and N the number of measurements or rows in data, in our case the number of rows.

2. We vary the pointwise error x until

   P^*[q_x(t_i) \le S(t_i)^*_b - S(t_i) \le q_{1-x}(t_i) \forall i = 1,\dots, N] \approx 1-\alpha.

   In other words, until the ratio of bootstrap curves that have all their points within [q_x(t_i), q_{1-x}(t_i)] is approximately 1-\alpha.

3. We define

   I_n(t_i) = [S(t_i) + q_x(t_i), S(t_i) + q_{1-x}(t_i)] \forall i = 1,\dots, N

   the confidence bounds. Then {I_n(t_i); i = 1,\dots, N} is a consistent simultaneous confidence band of level 1-\alpha.

Value

A data frame containing the upper confidence bound, the lower confidence bound, and the time point corresponding to the bounds.

References

Bühlmann, P. (1998). Sieve Bootstrap for Smoothing in Nonstationary Time Series. The Annals of Statistics, 26(1), 48-83.