Combination of Multi Level Forecasts

  • Silvia RiedelEmail author
  • Bogdan Gabrys


This paper provides a discussion of the effects of different multi-level learning approaches on the resulting out of sample forecast errors in the case of difficult real-world forecasting problems with large noise terms in the training data, frequently occurring structural breaks and quickly changing environments. In order to benefit from the advantages of learning on different aggregation levels and to reduce the risks of high noise terms on low level predictions and overgeneralization on higher levels, various approaches of using information at different levels are analysed in relation to their effects on the bias, variance and Bayes error components proposed by James and Hastie. We provide an extension of this decomposition for the multi-level case. An extensive analysis is also carried out answering the question of why the combination of predictions using information learned at different levels constitutes a significantly better approach in comparison to using only the predictions generated at one of the levels or other multi-level approaches. Additionally we argue why multi-level combinations should be used in addition to thick modelling and the use of different function spaces. Significant forecast improvements have been obtained when using the proposed multi-level combination approaches.


multi level forecasting forecast combination bias variance Bayes error decomposition  revenue management 


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  1. 1.
    G. Fliedner, “Hierarchical Forecasting: Issues and Use Guidelines,” Ind. Manage. Data Syst., vol. 1, 2001, pp. 5–12.CrossRefGoogle Scholar
  2. 2.
    G. James and T. Hastie, “Generalisations of the Bias/Variance Decomposition for Prediction Error”, technical report,, 1996.
  3. 3.
    T. D. Russell and E. E. Adam, “An Empirical Evaluation of Alternative Forecast Combinations,” Eur. J. Oper. Res. Econ., vol. 33, 1987, pp. 1267–1276.Google Scholar
  4. 4.
    L. M. De Menezes et al., “Review of Guidelines for the Use of Combined Forecasts,” Manage. Sci., vol. 120, 2000, pp. 190–204.zbMATHGoogle Scholar
  5. 5.
    A. G. Timmermann, “Forecast Combinations”, Discussion paper no. 5361,, 2005.
  6. 6.
    S. Makridakis et al., “The m2 Competition: A Real-Time Judgementally Based Forecasting Study”, Int. J. Forecast., vol. 9, 1993, pp. 5–22.Google Scholar
  7. 7.
    J. M. Bates and C. W. J. Granger, “The Combination of Forecasts,” Operations Research Quarterly, vol. 20, 1969, pp. 451–468.CrossRefGoogle Scholar
  8. 8.
    C. W. J. Granger and R. Ramanathan, “Improved Methods of Forecasting,” J. Forecast., vol. 3, 1984, pp. 197–204.CrossRefGoogle Scholar
  9. 9.
    E. W. Bunn, “Statistical Efficiency on the Linear Combination of Forecasts,” Int. J. Forecast., vol. 1, 1985, pp. 151–163.CrossRefzbMATHGoogle Scholar
  10. 10.
    J. V. Hansen, “Combining Predictors. Meta Machine Learning Methods and Bias/Variance and Ambiguity Decompositions,” Ph.D. dissertation, 2000.Google Scholar
  11. 11.
    S. Geman, E. Bienenstock, and R. Doursat, “Neural Networks and the Bias- Variance Dilemma,” Neural Comput., vol. 4, no. 1, 1992, pp. 1–58.CrossRefGoogle Scholar
  12. 12.
    R. G. Cross, “Revenue Management,” Broadway Books, 1997.Google Scholar
  13. 13.
    McGill and van Ryzin, “Revenue Management: Research Overview and Prospects,” Transp. Sci., vol. 33, no. 4, 1999.Google Scholar
  14. 14.
    C. W. J. Granger and Y. Jeon, “Thick Modelling,” Econometric Modelling, vol. 21, 2004, pp. 323–334.CrossRefGoogle Scholar
  15. 15.
    M. Aiolfo and C. A. Favero, “Model Uncertainty, Thick Modelling and the Predictability of Stock Returns,” J. Forecast., vol. 24, 2005, pp. 233–254.MathSciNetCrossRefGoogle Scholar
  16. 16.
    S. Riedel and B. Gabrys, “Hierarchical Multilevel Approaches of Forecast Combination,” Proceedings of the OR 2004 conference, The Netherlands, 2004.Google Scholar
  17. 17.
    S. Riedel and B. Gabrys, “Evolving Multilevel Forecast Combination Models—An Experimental Study,” Proceedings of NiSIS 2005 Symposium, Albufeira, Portugal, 2005.Google Scholar
  18. 18.
    S. Riedel and B. Gabrys, “Adaptive Mechanisms in an Airline Ticket Demand Forecasting System,” Proceedings of the EUNITE 2003 conference, Oulu, Finland, 2003.Google Scholar
  19. 19.
    R. Neuling, S. Riedel, and K.-U. Kalka, “New Approaches to Origin and Destination and No-show Forecasting: Excavating the Passenger Name Records Treasure,” Journal of Revenue and Pricing Management, vol. 3, no. 1, 2004, pp. 62–72.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Lufthansa Systems Berlin GmbHBerlinGermany
  2. 2.Computational Intelligence Research Group, School of Design, Engineering and Computing Bournemouth UniversityPooleUK

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