Cross-temporal Joint Block Bootstrap

Joint block bootstrap for generating probabilistic base forecasts that take into account the correlation between variables at different temporal aggregation orders (Girolimetto et al. 2023).

Usage

ctboot(model_list, boot_size, agg_order, block_size = 1, seed = NULL,
       xreg = NULL, ...)

Arguments

model_list

A list of \(p\) elements, one for each temporal level ordered from the lowest frequency (most temporally aggregated) to the highest frequency. Each elements is a list with the \(n\) base forecasts models for each cross-sectional series. A simulate() function for each model has to be available and implemented according to the package. forecast, with the following mandatory parameters: object, innov, future, and nsim.

boot_size

The number of bootstrap replicates.

agg_order

Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation, \(m\)), or a vector representing a subset of \(p\) factors of \(m\).

block_size

Block size of the bootstrap, which is typically equivalent to the forecast horizon for the most temporally aggregated series.

seed

An integer seed.

xreg

An optional 3-d numeric array of dimensions (\(n \times \text{boot\_size}(k^\ast+m) \times N_{xreg}\)) containing the new values of xreg to be used for forecasting ordered from the lowest frequency to the highest frequency (columns) for each variable (rows). It can contain NAs.

...

Additional arguments for the simulate() function.

Value

A list with \(\text{boot\_size}\) matrix of size (\(n\times(k^\ast+m)\text{block\_size}\)).

References

Girolimetto, D., Athanasopoulos, G., Di Fonzo, T. and Hyndman, R.J. (2024), Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues. International Journal of Forecasting, 40, 3, 1134-1151. doi:10.1016/j.ijforecast.2023.10.003

See also

Bootstrap samples: csboot(), teboot()

Cross-temporal framework: ctbu(), ctcov(), ctlcc(), ctmo(), ctmvn(), ctrec(), ctsmp(), cttd(), cttools(), iterec(), tcsrec()

Examples

set.seed(123)

# Minimal example functions: each "model" stores Gaussian residuals;
# simulate() draws new innovations (innov=NULL) or uses the supplied
# ones (innov given).
simple_model <- function(res) {
  structure(list(residuals = res, sigma = sd(res)), class = "simple_model")
}
simulate.simple_model <- function(object, nsim = 1, innov = NULL,
                                  future = TRUE, ...) {
  if (is.null(innov)) {
    rnorm(nsim, mean = 0, sd = object$sigma)
  } else {
    as.numeric(innov)[seq_len(nsim)]
  }
}

# Cross-temporal hierarchy:
# - cross-sectional: Z = X + Y  => n = 3
# - temporal: annual-quarterly  => m = 4, kset = c(4, 2, 1), p = 3 levels
n <- 3
m <- 4
kset <- c(4, 2, 1)
n_obs_per_k <- 40 / kset

# Nested list: outer level = p temporal aggregation orders (low -> high
# frequency), inner level = n cross-sectional series.
model_list <- lapply(seq_along(kset), function(i) {
  lapply(seq_len(n), function(j) {
    simple_model(rnorm(n_obs_per_k[i]))
  })
})

# Joint block bootstrap: 50 replicates, block_size = 1
boot <- ctboot(model_list = model_list, boot_size = 50, agg_order  = m,
               block_size = 1, seed = 1)
#> Error in UseMethod("simulate"): no applicable method for 'simulate' applied to an object of class "simple_model"