Efficient search strategies identify hierarchical

For example, we have found that bias adjustments increase computational cost by about 0. Regarding the combination of the methods, the additional cost depends strongly on the computation time required for individually estimating the combined methods. In this regard, these strategies transform temporal hierarchies into a truly efficient forecasting solution, capable of producing fast, yet highly accurate forecasts.

Temporal hierarchies have been proven an effective solution for improving the performance of traditional forecasting methods, supporting at the same time aligned decisions at different planning horizons. The accuracy, robustness, coherency and uncertainty mitigation they imply, have made them a popular forecasting framework, highlighting the potential benefits of multiple temporal aggregation in decision making.

Having examined the limitations of the existing framework, this study aims at further improving the performance of temporal hierarchies by considering three different strategies: i Combining forecasts of multiple methods, ii applying bias adjustments and iii selectively implementing temporal hierarchies to avoid seasonal shrinkage.

These strategies can be applied either separately or simultaneously, being complements to the method considered for reconciling the base forecasts and completely independent from each other. Moreover, they are computationally cheap, utilising information which becomes directly available when generating the base forecasts or easy-to-compute derivatives of the forecasting process. The results show that replacing base forecasts coming from a single method with combinations of multiple methods has a great positive impact on forecasting performance across all aggregation levels, especially when the individual methods used are inaccurate or biased.

Similarly, mitigating the bias of the base forecasts leads to improved forecasting performance, being more significant at the higher levels of the hierarchy where historical data are typically scarce. Finally, selectively choosing between temporal hierarchies and traditional bottom-up modelling is also beneficial, allowing the better handling of highly-seasonal series for which the seasonal component is excessively damped by temporal hierarchies. The strategies proposed in this study indicate that temporal hierarchies are a useful, yet generalised framework for combining forecasts of different temporal levels that can be further expanded to provide even more accurate and robust results.

This work introduces such effective expansions, outlining some interesting ways towards that direction. Even if this study focuses on improving the performance of temporal hierarchies, the first two strategies presented could be equally considered for cross-sectional hierarchies. Future research could focus towards that direction. Finally, our work focuses on the forecasting performance of the point forecasts and it would be interesting to see if the results generalise for the case of probabilistic forecasts [ 5 ]. Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field.

Abstract Temporal hierarchies have been widely used during the past few years as they are capable to provide more accurate coherent forecasts at different planning horizons.

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Funding: The authors received no specific funding for this work. In summary, our contribution is threefold: We evaluate the performance of combination across methods to produce the base hierarchical forecasts and we contrast any improvements from forecast combination across methods with the improvements of forecast combination across aggregation levels. We consider the case of forecast bias and we empirically bias-adjust the base forecast prior to hierarchical reconciliation.

Bias adjustment of the base forecasts is expected to lead to less biased final reconciled forecasts but also to improve their accuracy. We explore the effect of seasonal shrinkage and selectively apply temporal hierarchies for forecasting based on the seasonal significance of the original series, as well as that of the rest of the temporal levels. Selectively applying temporal hierarchies thus, avoiding excess seasonal shrinkage can potentially improve forecasting accuracy by properly capturing the seasonality component.

Download: PPT. The bias-variance decomposition, according to which the Mean Square Error MSE is decomposed into a bias B and a variance V term, is a fundamental concept in forecasting [ 45 ], expressed as 2 The bias represents the consistent distance observed between the forecasts and the true values. In order to deal with this problem, this last strategy involves an heuristic rule for avoiding seasonal shrinkage, as follows: The existence of seasonality is examined across all aggregation levels. If the series is not seasonal in the original frequency, then temporal hierarchies are applied.

If the series is seasonal in the original frequency, then temporal hierarchies are applied only if at least half of the aggregated levels are seasonal. Otherwise, the base forecasts of the lowest level of the hierarchy are used for extrapolation, being reconciled for the rest of the levels through the bottom-up method. Then, in order for a series to be classified as seasonal, at least two of the following must be true: Eq 5 is true.

Table 1. Forecasting performance in terms of accuracy for the 2, monthly series of the M and M3 competitions. Table 2. Forecasting performance in terms of bias for the 2, monthly series of the M and M3 competitions.

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Fig 3. Fig 4. Forecasting performance improvements reported for using a combination of forecasts instead of individual ones. Fig 5. Forecasting performance improvements reported for applying additive bias adjustments instead of original WLS S reconciliation. Fig 6. Forecasting performance improvements reported for selectively applying temporal hierarchies to avoid seasonal shrinkage. 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. References 1.

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