Improving the forecasting performance of temporal hierarchies

Autoři: Evangelos Spiliotis aff001;  Fotios Petropoulos aff002;  Vassilios Assimakopoulos aff001
Působiště autorů: Forecasting and Strategy Unit, School of Electrical and Computer Engineering, National Technical University of Athens, Zografou, Greece aff001;  School of Management, University of Bath, Bath, United Kingdom aff002
Vyšlo v časopise: PLoS ONE 14(10)
Kategorie: Research Article


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. However, they still display some limitations, being mainly subject to the forecasting methods used for generating the base forecasts and the particularities of the examined series. This paper deals with such limitations 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. The proposed 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. Their effect is evaluated using the monthly series of the M and M3 competitions. The results are very promising, displaying lots of potential for improving the performance of temporal hierarchies, both in terms of accuracy and bias.

Klíčová slova:

Covariance – Decision making – Machine learning algorithms – Mathematical functions – Seasons – Extrapolation


1. Hyndman RJ, Ahmed RA, Athanasopoulos G, Shang HL. Optimal combination forecasts for hierarchical time series. Computational Statistics & Data Analysis. 2011;55(9):2579–2589. doi: 10.1016/j.csda.2011.03.006

2. Wickramasuriya SL, Athanasopoulos G, Hyndman RJ. Optimal Forecast Reconciliation for Hierarchical and Grouped Time Series Through Trace Minimization. Journal of the American Statistical Association. 2018; p. 1–16.

3. Athanasopoulos G, Hyndman RJ, Kourentzes N, Petropoulos F. Forecasting with temporal hierarchies. European Journal of Operational Research. 2017;262(1):60–74. doi: 10.1016/j.ejor.2017.02.046

4. Athanasopoulos G, Ahmed RA, Hyndman RJ. Hierarchical forecasts for Australian domestic tourism. International Journal of Forecasting. 2009;25(1):146–166. doi: 10.1016/j.ijforecast.2008.07.004

5. Jeon J, Panagiotelis A, Petropoulos F. Probabilistic forecast reconciliation with applications to wind power and electric load. European Journal of Operational Research. 2019;279(2):364–379. doi: 10.1016/j.ejor.2019.05.020

6. Lütkepohl H. Forecasting contemporaneously aggregated vector ARMA processes. Journal of Business & Economic Statistics. 1984;2(3):201–214. doi: 10.1080/07350015.1984.10509388

7. Gross CW, Sohl JE. Disaggregation methods to expedite product line forecasting. Journal of Forecasting. 1990;9(3):233–254. doi: 10.1002/for.3980090304

8. Dangerfield BJ, Morris JS. Top-down or bottom-up: Aggregate versus disaggregate extrapolations. International Journal of Forecasting. 1992;8(2):233–241. doi: 10.1016/0169-2070(92)90121-O

9. Fliedner G. An investigation of aggregate variable time series forecast strategies with specific subaggregate time series statistical correlation. Computers & Operations Research. 1999;26(10):1133–1149. doi: 10.1016/S0305-0548(99)00017-9

10. Zellner A, Tobias J. A note on aggregation, disaggregation and forecasting performance. Journal of Forecasting. 2000;19(5):457–465. doi: 10.1002/1099-131X(200009)19:5%3C457::AID-FOR761%3E3.0.CO;2-6

11. Hyndman RJ, Lee AJ, Wang E. Fast computation of reconciled forecasts for hierarchical and grouped time series. Computational Statistics & Data Analysis. 2016;97:16–32. doi: 10.1016/j.csda.2015.11.007

12. Abouarghoub W, Nomikos NK, Petropoulos F. On reconciling macro and micro energy transport forecasts for strategic decision making in the tanker industry. Transportation Research Part E: Logistics and Transportation Review. 2018;113:225–238. doi: 10.1016/j.tre.2017.10.012

13. Andrawis RR, Atiya AF, El-Shishiny H. Combination of long term and short term forecasts, with application to tourism demand forecasting. International Journal of Forecasting. 2011;27(3):870–886. doi: 10.1016/j.ijforecast.2010.05.019

14. Kourentzes N, Petropoulos F, Trapero JR. Improving forecasting by estimating time series structural components across multiple frequencies. International Journal of Forecasting. 2014;30(2):291–302. doi: 10.1016/j.ijforecast.2013.09.006

15. Petropoulos F, Kourentzes N. Improving forecasting via multiple temporal aggregation. Foresight: The International Journal of Applied Forecasting. 2014;34:12–17.

16. Petropoulos F, Kourentzes N. Forecast combinations for intermittent demand. The Journal of the Operational Research Society. 2015;66(6):914–924. doi: 10.1057/jors.2014.62

17. Kourentzes N, Petropoulos F. Forecasting with multivariate temporal aggregation: The case of promotional modelling. International Journal of Production Economics. 2016;181, Part A:145–153. doi: 10.1016/j.ijpe.2015.09.011

18. Kourentzes N, Rostami-Tabar B, Barrow DK. Demand forecasting by temporal aggregation: Using optimal or multiple aggregation levels? Journal of Business Research. 2017;78(Supplement C):1–9. doi: 10.1016/j.jbusres.2017.04.016

19. Nystrup P, Lindström E, Pinson P, Madsen H. Temporal hierarchies with autocorrelation for load forecasting. European Journal of Operational Research. 2019.

20. Spiliotis E, Petropoulos F, Kourentzes N, Assimakopoulos V. Cross-temporal aggregation: Improving the forecast accuracy of hierarchical electricity consumption. Forecasting and Strategy Unit Working Paper. 2018;1/18:1.

21. Kourentzes N, Athanasopoulos G. Cross-temporal coherent forecasts for Australian tourism. Annals of Tourism Research. 2019;forthcoming. doi: 10.1016/j.annals.2019.02.001

22. Yagli GM, Yang D, Srinivasan D. Reconciling solar forecasts: Sequential reconciliation. Solar Energy. 2019;179:391–397. doi: 10.1016/j.solener.2018.12.075

23. Ben Taieb S, Taylor JW, Hyndman RJ. Coherent probabilistic forecasts for hierarchical time series. Proceedings of the 34th International Conference on Machine Learning. 2017;70:3348–3357.

24. Makridakis S, Winkler RL. Averages of Forecasts: Some Empirical Results. Management science. 1983;29(9):987–996. doi: 10.1287/mnsc.29.9.987

25. Clemen RT. Combining forecasts: A review and annotated bibliography. International journal of forecasting. 1989;5(4):559–583. doi: 10.1016/0169-2070(89)90012-5

26. Timmermann A. Forecast Combinations. Handbook of Economic Forecasting. 2006;1:135–196. doi: 10.1016/S1574-0706(05)01004-9

27. Hibon M, Evgeniou T. To combine or not to combine: selecting among forecasts and their combinations. International journal of forecasting. 2005;21:15–24. doi: 10.1016/j.ijforecast.2004.05.002

28. Jose VRR, Winkler RL. Simple robust averages of forecasts: Some empirical results. International journal of forecasting. 2008;24:163–169. doi: 10.1016/j.ijforecast.2007.06.001

29. Petropoulos F, Svetunkov I. A Simple Combination of Univariate Models. International journal of forecasting. 2019;forthcoming. doi: 10.1016/j.ijforecast.2019.01.006

30. Bates JM, Granger CWJ. The Combination of Forecasts. Journal of the Operational Research Society. 1969;20(4):451–468. doi: 10.1057/jors.1969.103

31. Clemen RT. Combining forecasts: A review and annotated bibliography. International Journal of Forecasting. 1989;5(4):559–583.

32. Makridakis S, Winkler RL. Averages of Forecasts: Some Empirical Results. Management Science. 1983;29(9):987–996. doi: 10.1287/mnsc.29.9.987

33. Petropoulos F, Hyndman RJ, Bergmeir C. Exploring the sources of uncertainty: Why does bagging for time series forecasting work? European Journal of Operational Research. 2018;268(2):545–554.

34. Claeskens G, Magnus JR, Vasnev AL, Wang W. The forecast combination puzzle: A simple theoretical explanation. International Journal of Forecasting. 2016;32(3):754–762.

35. Smith J, Wallis KF. A Simple Explanation of the Forecast Combination Puzzle*. Oxford Bulletin of Economics and Statistics. 2009;71(3):331–355. doi: 10.1111/j.1468-0084.2008.00541.x

36. Chan F, Pauwels LL. Some theoretical results on forecast combinations. International Journal of Forecasting. 2018;34(1):64–74.

37. Makridakis S, Spiliotis E, Assimakopoulos V. The M4 Competition: 100,000 time series and 61 forecasting methods. International Journal of Forecasting. 2019;forthcoming. doi: 10.1016/j.ijforecast.2019.04.014

38. Lichtendahl C, Winkler B. Why Do Some Combinations Perform Better Than Others? International Journal of Forecasting. 2019;forthcoming. doi: 10.1016/j.ijforecast.2019.03.027

39. Kourentzes N, Barrow DK, Crone SF. Neural network ensemble operators for time series forecasting. Expert Systems with Applications. 2014;41(9):4235–4244.

40. Jose VRR, Grushka-Cockayne Y, Lichtendahl KC. Trimmed Opinion Pools and the Crowd’s Calibration Problem. Management Science. 2014;60(2):463–475. doi: 10.1287/mnsc.2013.1781

41. Grushka-Cockayne Y, Jose VRR, Lichtendahl KC. Ensembles of Overfit and Overconfident Forecasts. Management Science. 2017;63(4):1110–1130. doi: 10.1287/mnsc.2015.2389

42. Montero-Manso P, Athanasopoulos G, Hyndman RJ, Talagala TS. FFORMA: Feature-based Forecast Model Averaging. International Journal of Forecasting. 2019;forthcoming.

43. Nikolopoulos K, Petropoulos F. Forecasting for big data: Does suboptimality matter? Computers & Operations Research. 2018;98:322—329.

44. Seaman B. Considerations of a retail forecasting practitioner. International Journal of Forecasting. 2018;34(4):822–829.

45. Taieb SB, Atiya AF. A Bias and Variance Analysis for Multistep-Ahead Time Series Forecasting. IEEE Transactions on Neural Networks and Learning Systems. 2016;27(1):62–76. doi: 10.1109/TNNLS.2015.2411629 25807572

46. den Broeke MV, Baets SD, Vereecke A, Baecke P, Vanderheyden K. Judgmental forecast adjustments over different time horizons. Omega. 2018.

47. Arvan M, Fahimnia B, Reisi M, Siemsen E. Integrating human judgement into quantitative forecasting methods: A review. Omega. 2019;86:237–252.

48. Kim H, Durmaz N. Bias correction and out-of-sample forecast accuracy. International Journal of Forecasting. 2012;28(3):575–586.

49. Hyndman RJ, Athanasopoulos G. Forecasting: principles and practice, 2nd edition. OTexts: Melbourne, Australia; 2018. Available from:

50. Petropoulos F, Wang X, Disney SM. The inventory performance of forecasting methods: Evidence from the M3 competition data. International Journal of Forecasting. 2019;35(1):251–265. doi: 10.1016/j.ijforecast.2018.01.004

51. Miller DM, Williams D. Shrinkage estimators of time series seasonal factors and their effect on forecasting accuracy. International Journal of Forecasting. 2003;19(4):669–684.

52. Spiliotis E, Assimakopoulos V, Nikolopoulos K. Forecasting with a hybrid method utilizing data smoothing, a variation of the Theta method and shrinkage of seasonal factors. International Journal of Production Economics. 2019;209:92–102.

53. Assimakopoulos V, Nikolopoulos K. The Theta model: a decomposition approach to forecasting. International Journal of Forecasting. 2000;16(4):521–530. doi: 10.1016/S0169-2070(00)00066-2

54. Fiorucci JA, Pellegrini TR, Louzada F, Petropoulos F, Koehler AB. Models for optimising the theta method and their relationship to state space models. International Journal of Forecasting. 2016;32(4):1151–1161. doi: 10.1016/j.ijforecast.2016.02.005

55. Hyndman RJ, Koehler AB, Snyder RD, Grose S. A state space framework for automatic forecasting using exponential smoothing methods. International Journal of Forecasting. 2002;18(3):439–454.

56. Hyndman R, Khandakar Y. Automatic Time Series Forecasting: The forecast Package for R. Journal of Statistical Software, Articles. 2008;27(3):1–22.

57. Makridakis S, Andersen A, Carbone R, Fildes R, Hibon M, Lewandowski R, et al. The accuracy of extrapolation (time series) methods: Results of a forecasting competition. Journal of Forecasting. 1982;1(2):111–153. doi: 10.1002/for.3980010202

58. Makridakis S, Hibon M. The M3-Competition: results, conclusions and implications. International Journal of Forecasting. 2000;16(4):451–476.

59. Hyndman R, Athanasopoulos G, Bergmeir C, Caceres G, Chhay L, O’Hara-Wild M, et al. forecast: Forecasting functions for time series and linear models; 2018. Available from:

60. Hyndman RJ, Koehler AB. Another look at measures of forecast accuracy. International Journal of Forecasting. 2006;22(4):679–688.

61. Goodwin P, Lawton R. On the asymmetry of the symmetric MAPE. International Journal of Forecasting. 1999;15(4):405–408.

62. Koning AJ, Franses PH, Hibon M, Stekler HO. The M3 competition: Statistical tests of the results. International Journal of Forecasting. 2005;21(3):397–409. doi: 10.1016/j.ijforecast.2004.10.003

Článek vyšel v časopise


2019 Číslo 10
Nejčtenější tento týden