Advertisement

Game-Theoretically Optimal Reconciliation of Contemporaneous Hierarchical Time Series Forecasts

  • Tim van ErvenEmail author
  • Jairo Cugliari
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 217)

Abstract

In hierarchical time series (HTS) forecasting, the hierarchical relation between multiple time series is exploited to make better forecasts. This hierarchical relation implies one or more aggregate consistency constraints that the series are known to satisfy. Many existing approaches, like for example bottom-up or top-down forecasting, therefore attempt to achieve this goal in a way that guarantees that the forecasts will also be aggregate consistent. We propose to split the problem of HTS into two independent steps: first one comes up with the best possible forecasts for the time series without worrying about aggregate consistency; and then a reconciliation procedure is used to make the forecasts aggregate consistent. We introduce a Game-Theoretically OPtimal (GTOP) reconciliation method, which is guaranteed to only improve any given set of forecasts. This opens up new possibilities for constructing the forecasts. For example, it is not necessary to assume that bottom-level forecasts are unbiased, and aggregate forecasts may be constructed by regressing both on bottom-level forecasts and on other covariates that may only be available at the aggregate level. We illustrate the benefits of our approach both on simulated data and on real electricity consumption data.

Keywords

Prediction Interval Bregman Divergence Regional Forecast Initial Forecast Bregman Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors would like to thank Mesrob Ohannessian for useful discussions, which led to the closed-form solution for the GTOP predictions in Example 1. We also thank two anonymous referees for useful suggestions to improve the presentation. This work was supported in part by NWO Rubicon grant 680-50-1112.

References

  1. 1.
    Antoniadis, A., Brossat, X., Cugliari, J., & Poggi, J.-M. (2012). Prévision d’un processus à valeurs fonctionnelles en présence de non stationnarités. Application à la consommation d’électricité. Journal de la Société Française de Statistique, 153(2), 52–78.MathSciNetGoogle Scholar
  2. 2.
    Antoniadis, A., Brossat, X., Cugliari, J., & Poggi, J. M. (2013). Une approche fonctionnelle pour la prévision non-paramétrique de la consommation d’électricité. Technical report oai:hal.archives-ouvertes.fr:hal-00814530, Hal, Avril 2013. http://hal.archives-ouvertes.fr/hal-00814530.
  3. 3.
    Borges, C. E., Penya, Y. K., & Fernández, I. (2013). Evaluating combined load forecasting in large power systems and smart grids. IEEE Transactions on Industrial Informatics, 9(3), 1570–1577.CrossRefGoogle Scholar
  4. 4.
    Byron, R. P. (1978). The estimation of large social account matrices. Journal of the Royal Statistical Society, Series A, 141, 359–367.zbMATHCrossRefGoogle Scholar
  5. 5.
    Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, learning, and games. Cambridge: Cambridge University Press.zbMATHCrossRefGoogle Scholar
  6. 6.
    Chen, B. (2006). A balanced system of industry accounts for the U.S. and structural distribution of statistical discrepancy. Technical report, Bureau of Economic Analysis. http://www.bea.gov/papers/pdf/reconciliation_wp.pdf.
  7. 7.
    Fliedner, G. (1999). An investigation of aggregate variable time series forecast strategies with specific subaggregate time series statistical correlation. Computers & Operations Research, 26, 1133–1149.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Granger, C. W. J. (1988). Aggregation of time series variables—A survey. Discussion paper 1, Federal Reserve Bank of Minneapolis, Institute for Empirical Macroeconomics. http://www.minneapolisfed.org/research/DP/DP1.pdf.
  9. 9.
    Hazan, E., Agarwal, A., & Kale, S. (2007). Logarithmic regret algorithms for online convex optimization. Machine Learning, 69(2–3), 169–192.CrossRefGoogle Scholar
  10. 10.
    Hong, T., Pinson, P., & Fan, S. (2014). Global energy forecasting competition 2012. International Journal of Forecasting, 30, 357–363.CrossRefGoogle Scholar
  11. 11.
    Hyndman, R. J., Ahmed, R. A., Athanasopoulos, G., & Shang, H. L. (2011). Optimal combination forecasts for hierarchical time series. Computational Statistics and Data Analysis, 55, 2579–2589.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lobo, M. S., Vandenberghe, L., Boyd, S., & Lebret, H. (1998). Applications of second-order cone programming. Linear Algebra and its Applications, 284(1–3), 193–228.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Lütkepohl, H. (2009). Forecasting aggregated time series variables: A survey. Working paper EUI ECO: 2009/17, European University Institute. http://hdl.handle.net/1814/11256.
  14. 14.
    MacKinnon, J. G., & White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29, 305–325.CrossRefGoogle Scholar
  15. 15.
    Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press.zbMATHGoogle Scholar
  16. 16.
    Stone, R., Champernowne, D. G., & Meade, J. E. (1942). The precision of national income estimates. The Review of Economic Studies, 9(2), 111–125.CrossRefGoogle Scholar
  17. 17.
    Tibshirani, R. (1996). Regression shrinkage and selection via the LASSO. Journal of the Royal Statistical Society, Series B, 58(1), 267–288.zbMATHMathSciNetGoogle Scholar
  18. 18.
    White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817–838.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Université Paris-SudOrsay CedexFrance
  2. 2.INRIA Saclay – Ile de France, Select teamUniversité Paris-SudOrsay CedexFrance
  3. 3.Laboratoire ERICUniversité Lumière Lyon2Bron CedexFrance

Personalised recommendations