Top-down forecast reconciliation for genuine hierarchical/grouped time series

Top-down forecast reconciliation for genuine hierarchical/grouped time series, where the forecast of a `Total' (top-level series, expected to be positive) is disaggregated according to a proportional scheme given by a vector of proportions (weights). Besides the fulfillment of any aggregation constraint, the top-down reconciled forecasts should respect two main properties:

the top-level value remains unchanged;
all the bottom time series reconciled forecasts are non-negative.

The top-down procedure is extended to deal with both temporal and cross-temporal cases. Since this is a post-forecasting function, the vector of weights must be given in input by the user, and is not calculated automatically (see Examples).

tdrec(topf, C, m, weights)

Arguments

topf: ( $h \times 1$ ) vector of the top-level base forecast to be disaggregated; $h$ is the forecast horizon (for the lowest temporal aggregation order in temporal and cross-temporal cases).
C: ( $n_a \times n_b$ ) cross-sectional (contemporaneous) matrix mapping the $n_b$ bottom level series into the $n_a$ higher level ones.
m: Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation, $m$ ), or a subset of the $p$ factors of $m$ .
weights: vector of weights to be used to disaggregate topf: ( $n_b \times h$ ) matrix in the cross-sectional framework; ( $m \times h$ ) matrix in the temporal framework; ( $n_b m \times h$ ) matrix in the cross-temporal framework.

Value

The function returns an ( $h \times n$ ) matrix of cross-sectionally reconciled forecasts, or an ( $h(k^\ast + m) \times 1$ ) vector of top-down temporally reconciled forecasts, or an ( $n \times h (k^\ast + m)$ ) matrix of top-down cross-temporally reconciled forecasts.

Details

Fix $h = 1$ , then $\widetilde{\mathbf{y}} = \mathbf{S}\mathbf{w}\widehat{a}_1$ where $\widetilde{\mathbf{y}}$ is the vector of reconciled forecasts, $\mathbf{S}$ is the summing matrix (whose pattern depends on which type of reconciliation is being performed), $\mathbf{w}$ is the vector of weights, and $\widehat{a}_1$ is the top-level value to be disaggregated.

References

Athanasopoulos, G., Ahmed, R.A., Hyndman, R.J. (2009), Hierarchical forecasts for Australian domestic tourism, International Journal of Forecasting, 25, 1, 146–166.

Examples

data(FoReco_data)
### CROSS-SECTIONAL TOP-DOWN RECONCILIATION
# Cross sectional aggregation matrix
C <- FoReco_data$C
# monthly base forecasts
mbase <- FoReco2matrix(FoReco_data$base, m = 12)$k1
obs_1 <- FoReco_data$obs$k1
# average historical proportions
props <- colMeans(obs_1[1:168,-c(1:3)]/obs_1[1:168,1])
cs_td <- tdrec(topf = mbase[,1], C = C, weights = props)

### TEMPORAL TOP-DOWN RECONCILIATION
# top ts base forecasts ([lowest_freq' ...  highest_freq']')
top_obs12 <- FoReco_data$obs$k12[1:14,1]
bts_obs1 <- FoReco_data$obs$k1[1:168,1]
# average historical proportions
props <- colMeans(matrix(bts_obs1, ncol = 12, byrow = TRUE)/top_obs12)
topbase <- FoReco_data$base[1, 1]
t_td <- tdrec(topf = topbase, m = 12, weights = props)

### CROSS-TEMPORAL TOP-DOWN RECONCILIATION
top_obs <- FoReco_data$obs$k12[1:14,1]
bts_obs <- FoReco_data$obs$k1[1:168,-c(1:3)]
bts_obs <- lapply(1:5, function(x) matrix(bts_obs[,x], nrow=14, byrow = TRUE))
bts_obs <- do.call(cbind, bts_obs)
# average historical proportions
props <- colMeans(bts_obs/top_obs)
ct_td <- tdrec(topf = topbase, m = 12, C = C, weights = props)