Fig 1.
A cross-sectional hierarchy.
Fig 2.
A temporal hierarchy.
Table 1.
Forecasting performance in terms of accuracy for the 2,045 monthly series of the M and M3 competitions.
The results are averaged across all series, both per aggregation level and across all levels. Additive (ADD), multiplicative (MUL) or none bias adjustments were considered. Results are also reported when temporal hierarchical forecasting is applied selectively to avoid seasonal shrinkage.
Table 2.
Forecasting performance in terms of bias for the 2,045 monthly series of the M and M3 competitions.
The results are averaged across all series, both per aggregation level and across all levels. Additive (ADD), multiplicative (MUL) or none bias adjustments were considered. Results are also reported when temporal hierarchical forecasting is applied selectively to avoid seasonal shrinkage.
Fig 3.
Forecasting performance improvements reported for applying WLSS reconciliation instead of BU.
The improvements (percentage differences) are estimated per aggregation level for the ETS (red), ARIMA (green) and COMB (blue) methods, both in terms of accuracy (MASE) and bias (ASME).
Fig 4.
Forecasting performance improvements reported for using a combination of forecasts instead of individual ones.
The improvements (percentage differences) are estimated per aggregation level, both in terms of accuracy (MASE) and bias (ASME). Comparisons of COMB are presented independently for ETS (red) and ARIMA (green), both for applying the BU (solid) and WLSS (dashed) reconciling methods.
Fig 5.
Forecasting performance improvements reported for applying additive bias adjustments instead of original WLSS reconciliation.
The improvements (percentage differences) are estimated per aggregation level for the ETS (red), ARIMA (green) and COMB (blue) methods, both in terms of accuracy (MASE) and bias (ASME). A similar graph is obtained for the case of multiplicative bias adjustments.
Fig 6.
Forecasting performance improvements reported for selectively applying temporal hierarchies to avoid seasonal shrinkage.
The improvements (percentage differences) are estimated per aggregation level for the ETS (red), ARIMA (green) and COMB (blue) methods, both in terms of accuracy (MASE) and bias (ASME). The WLSS estimator is used for reconciliation, either considering additive bias adjustments “BA” (dashed) or not (solid). A similar graph is obtained for the case of multiplicative bias adjustments.
Table 3.
Average performance improvements when each strategy is applied separately or in conjuction with other strategies.
Fig 7.
MCB significance tests for the three strategies considered.
The analysis is done using the average MASE reported for each series across all temporal levels. In all cases, the WLSS reconciliation method is considered. “COMB” is the combination of ETS and ARIMA forecasts (first strategy), “BA” is additive bias adjustments (second strategy), “SS” is the avoidance of seasonal shrinkage (third strategy) and “COMB AS” is the combination of the three strategies.