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Issue No.12 - December (2009 vol.21)
pp: 1753-1766
Silvia Riedel , Lufthansa Systems Berlin GmbH, Berlin
Bogdan Gabrys , Bournemouth University, Poole
In this paper, we provide a theoretical analysis of effects of applying different forecast diversification methods on the structure of the forecast error covariance matrices and decomposed forecast error components based on the bias-variance-Bayes error decomposition of James and Hastie. We express the "diversity” of different forecasts in relation to different error components and propose a measure in order to quantify it. We illustrate and discuss typical inhomogeneities frequently occurring in the forecast error covariance matrices and show that previously proposed pooling based only on error variances cannot fully exploit the complementary information present in a set of diverse forecasts to be combined. If covariance values could be reliably calculated, they could be taken into account during the pooling process. We study the difficult case in which covariance information cannot be measured properly and propose a novel simplified representation of the covariance matrix, which is only based on knowledge about the forecast generation process. Finally, we propose a new pooling approach that avoids inhomogeneities in the forecast error covariance matrix by considering the information contained in the simplified covariance representation and compare it with the error-variance-based pooling approach introduced by Aiolfi and Timmermann. Applying our approach more than once leads to the generation of multistep and multilevel forecast combination structures, which have generated significantly improved forecasts in our previous extensive experimental work; the summary of which is also provided.
Forecast combination, pooling, multilevel forecasting, thick modeling.
Silvia Riedel, Bogdan Gabrys, "Pooling for Combination of Multilevel Forecasts", IEEE Transactions on Knowledge & Data Engineering, vol.21, no. 12, pp. 1753-1766, December 2009, doi:10.1109/TKDE.2009.18
[1] J.G.D. Gooijer and R.J. Hyndman, “25 Years of Time Series Forecasting,” Int'l J. Forecasting, vol. 22, no. 3, pp. 443-473, 2006.
[2] A. Timmermann, “Forecast Combinations,” Handbook of Economic Forecasting, G. Elliott, C. Granger, and A. Timmermann, eds., pp.135-196, Elsevier, 2006.
[3] R.S. Tsai, Analysis of Financial Time Series. John Wiley & Sons, 2002.
[4] D. Ruta and B. Gabrys, “Neural Network Ensembles for Time Series Prediction,” Proc. Int'l Joint Conf. Neural Networks (IJCNN '07), pp. 1204-1209, 2007.
[5] V. Alarcon-Aquino and J.A. Barria, “Multiresolution FIR Neural Network Based Learning Algorithm Applied to Network Traffic Prediction,” IEEE Trans. Systems, Man and Cybernetics Part C: Applications and Rev., vol. 36, no. 2, pp. 208-220, Mar. 2006.
[6] Z. Vojinovic, V. Kecman, and R. Seidel, “A Data Mining Approach to Financial Time Series Modelling and Forecasting,” Int'l J. Intelligent Systems in Accounting, Finance & Management, vol. 10, no. 4, pp. 225-239, 2001.
[7] C.L. Giles, S. Lawrence, and A.C. Tsoi, “Noisy Time Series Prediction Using Recurrent Neural Networks and Grammatical Inference,” Machine Learning, vol. 44, no. 1, pp. 335-356, 2001.
[8] F.E.H. Tay and L.J. Cao, “Modified Support Vector Machines in Financial Time Series Forecasting,” Neurocomputing, vol. 48, no. 1, pp. 847-861, 2002.
[9] M. Casdagli, “Nonlinear Prediction of Chaotic Time Series,” Physica, vol. 35, pp. 335-356, 1989.
[10] S. Makridakis and M. Hibon, “The M3-Competition: Results, Conclusions and Implications,” Int'l J. Forecasting, vol. 16, no. 4, pp. 451-476, 2000.
[11] S. Riedel, “Forecast Combination in Revenue Management Demand Forecasting,” PhD thesis, Bournemouth Univ., 2008.
[12] S. Riedel and B. Gabrys, “Combination of Multi Level Forecasts,” J. of VLSI Signal Processing Systems, special issue on data fusion for medical, industrial, and environmental applications, vol. 49, no. 2, pp. 265-280, 2007.
[13] S. Riedel and B. Gabrys, “Dynamic Pooling for the Combination of Forecasts Generated Using Multi Level Learning,” Proc. Int'l Joint Conf. Neural Networks (IJCNN '07), pp. 454-459, 2007.
[14] C.W.J. Granger and Y. Jeon, “Thick Modeling,” Econometric Modeling, vol. 21, pp. 323-343, 2004.
[15] H. Goldstein, Multilevel Statistical Models, third ed. Oxford Univ. Press, 2003.
[16] S.W. Raudenbush and A.S. Bryk, Hierarchical Linear Models: Applications and Data Analysis Methods. Sage, 2002.
[17] T.A.B. Snijders and R.J. Bosker, Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling. Sage, 1999.
[18] T.D. Russell and E.E. Adam, “An Empirical Evaluation of Alternative Forecast Combinations,” European J. Operations Research, vol. 33, pp. 1267-1276, 1987.
[19] L.M. De Menezes et al., “Review of Guidelines for the Use of Combined Forecasts,” Management Science, vol. 120, pp. 190-204, 2000.
[20] J.M. Bates and C.W.J. Granger, “The Combination of Forecasts,” Operations Research Quarterly, vol. 20, pp. 451-468, 1969.
[21] C.W.J. Granger and R. Ramanathan, “Improved Methods of Forecasting,” J. Forecasting, vol. 3, pp. 197-204, 1984.
[22] E.W. Bunn, “Statistical Efficiency on the Linear Combination of Forecasts,” Int'l J. Forecasting, vol. 1, pp. 151-163, 1985.
[23] M. Aiolfi and A.G. Timmermann, “Persistence of Forecasting Performance and Conditional Combination Strategies,” J. Econometrics, vol. 135, pp. 31-53, 2006.
[24] S. Riedel and B. Gabrys, “Hierarchical Multilevel Approaches of Forecast Combination,” Proc. Operations Research Conf., 2004.
[25] S. Riedel and B. Gabrys, “Evolving Multilevel Forecast Combination Models—An Experimental Study,” Proc. Nature-Inspired Smart Information Systems Conf., 2005.
[26] G. James and T. Hastie, “Generalisations of the Bias/Variance Decomposition for Prediction Error,” technical report, www.stat., 1996.
[27] J.V. Hansen, “Combining Predictors. Meta Machine Learning Methods and Bias/Variance and Ambiguity Decompos,” PhD dissertation, Univ. of Aarhus, 2000.
[28] S. Geman, E. Bienenstock, and R. Doursat, “Neural Networks and the Bias-Variance Dilemma,” Neural Computation, vol. 4, pp. 1-58, 1992.
[29] J.S. Armstrong, “Combining Forecasts: The End of the Beginning or Beginning of the End,” Int'l J. Forecasting, vol. 5, pp. 585-588, 1989.
[30] D. Ruta and B. Gabrys, “Classifier Selection for Majority Voting,” J. Information Fusion, special issue on diversity in multiple classifier systems, vol. 6, pp. 63-81, 2005.
[31] J.I. McGill and G.J. van Ryzin, “Revenue Management: Research Overview and Prospects,” Transportation Science, vol. 33, pp. 233-256, 1999.
[32] S. Riedel and B. Gabrys, “Adaptive Mechanisms in an Airline Ticket Demand Forecasting System,” Proc. European Symp. Intelligent Technologies Conf., 2003.
[33] R. Neuling, S. Riedel, and K.-U. Kalka, “New Approaches to Origin and Destination and No-Show Forecasting: Excavating the Passenger Name Records Treasure,” J. Revenue and Pricing Management, vol. 3, pp. 62-72, 2004.
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