Cross-temporal probabilistic reconciliation (sample approach)

This function performs cross-temporal probabilistic forecast reconciliation using a sample-based approach (Girolimetto et al., 2024) for linearly constrained multiple time series observed across both cross-sectional and temporal dimensions (Di Fonzo and Girolimetto, 2023). Given a \((n \times h(k^\ast + m) \times L)\) array of simulated base forecast draws, ctsmp() applies a chosen FoReco cross-temporal reconciliation independently to each draw, enforcing coherence simultaneously over aggregation levels and across series. Typical choices for the reconciliation include optimal combination (ctrec) as well as top-down (cttd), middle-out (ctmo), bottom-up (ctbu), and level-conditional (ctlcc) approaches.

Usage

ctsmp(sample, agg_order, fun = ctrec, ...)

Arguments

sample: A (\(n \times h(k^\ast + m) \times L\)) numeric array containing the base forecasts samples to be reconciled; \(n\) is the total number of variables, \(m\) is the max. order of temporal aggregation, \(k^\ast\) is the sum of (a subset of) (\(p-1\)) factors of \(m\), excluding \(m\), \(h\) is the forecast horizon for the lowest frequency time series, and \(L\) is the sample size. The row identifies a time series, and the forecasts in each row are ordered from the lowest frequency (most temporally aggregated) to the highest frequency.
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 \(m\).
fun: A string specifying the reconciliation function to be used, as implemented in FoReco.
...: Arguments passed on to fun

Value

A distributional::dist_sample object.

References

Di Fonzo, T. and Girolimetto, D. (2023a), Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives, International Journal of Forecasting, 39, 1, 39-57. doi:10.1016/j.ijforecast.2021.08.004

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

Examples

set.seed(123)
A <- t(c(1,1)) # Aggregation matrix for Z = X + Y
m <- 4 # from quarterly to annual temporal aggregation

# (100 x 14) base forecasts sample matrix (simulated), m = 4, h = 2, n = 3
sample <- simplify2array(lapply(1:100, function(x){
  rbind(rnorm(14, rep(c(20, 10, 5), 2*c(1, 2, 4))),
        rnorm(14, rep(c(10, 5, 2.5), 2*c(1, 2, 4))),
        rnorm(14, rep(c(10, 5, 2.5), 2*c(1, 2, 4))))
}))
# (3 x 70) in-sample residuals matrix (simulated)
res <- rbind(rnorm(70), rnorm(70), rnorm(70))

# Top-down probabilistic reconciliation
reco_dist_td <- ctsmp(sample[1, 1:2, , drop = FALSE], agg_order = m,
                      agg_mat = A, fun = cttd, weights = matrix(runif(8), 2))

# Middle-out probabilistic reconciliation
reco_dist_mo <- ctsmp(sample[1, 3:6, , drop = FALSE], agg_order = m,
                      agg_mat = A, fun = ctmo, weights = matrix(runif(8), 2),
                      id_rows = 1, order = 2)
# Bottom-up probabilistic reconciliation
reco_dist_bu <- ctsmp(sample[-1,-c(1:6), ], agg_order = m, agg_mat = A,
                      fun = ctbu)

# Level conditional coherent probabilistic reconciliation
reco_dist_lcc <- ctsmp(sample, agg_order = m, agg_mat = A, fun = ctlcc)
#> Warning: Argument `res` is ignored.
#> ℹ When `res` is provided, `comb = 'bdshr'` is suggested.

# Optimal cross-sectional probabilistic reconciliation
reco_dist_opt <- ctsmp(sample, agg_order = m, agg_mat = A, res = res,
                       comb = "bdshr")