Abstract
Time series modeling and forecasting are essential in many domains of science and engineering. Extensive works in literature suggest that combining outputs of different forecasting methods substantially increases the overall accuracies as well as reduces the risk of model selection. The most popular method of forecasts combination is the weighted averaging of the constituent forecasts. The effectiveness of this method solely depends on appropriate selection of the combining weights. In this paper, we comprehensively evaluate a wide variety of benchmark weights selection techniques for linear combination of multiple forecasts in terms of their prediction accuracies. Nine real-world time series from different domains and five individual forecasting methods are used in our empirical work. A robust scheme is also suggested for fairly ranking the combination methods on the basis of their forecasting performances. Our study precisely demonstrates the relative strengths and weaknesses of various benchmark linear combination techniques which evidently can be of much practical importance.
Similar content being viewed by others
References
Aksu C, Gunter S (1992) An empirical analysis of the accuracy of SA, OLS, ERLS and NRLS combination forecasts. Int J Forecast 8(1): 27–43
Andrawis RR, Atiya AF, El-Shishiny H (2011) Forecast combinations of computational intelligence and linear models for the NN5 time series forecasting competition. Int J Forecast 27(3): 672–688
Armstrong JS (2001) Principles of forecasting: a handbook for researchers and practitioners. Kluwer Academic Publishers, Boston, USA
Bates JM, Granger CWJ (1969) Combination of forecasts. Oper Res Q 20(4): 451–468
Box GEP, Jenkins GM (1970) Time series analysis, forecasting and control, 3rd edn. Holden-Day, California
Bunn D (1975) A Bayesian approach to the linear combination of forecasts. Oper Res Q 26(2): 325–329
Chan CK, Kingsman BG, Wong H (2004) Determining when to update the weights in combined forecasts for product demand—an application of the CUSUM technique. Eur J Oper Res 153(3): 757–768
Chapelle O (2002) Support vector machines: introduction principles, adaptive tuning and prior knowledge. Ph.D. Thesis, University of Paris, France
Clemen RT (1989) Combining forecasts: a review and annotated bibliography. J Forecast 5(4): 559–583
De Gooijer JG, Hyndman RJ (2006) 25 years of time series forecasting. J Forecast 22(3): 443–473
De Menezes LM, Bunn DW, Taylor JW (2000) Review of guidelines for the use of combined forecasts. Eur J Oper Res 120(1): 190–204
Demuth H, Beale M, Hagan M (2010) Neural network toolbox user’s guide. The MathWorks, Natic
Frietas PSA, Rodrigues AJL (2006) Model combination in neural-based forecasting. Eur J Oper Res 173(3): 801–814
Hamzaçebi C, Akay D, Kutay F (2009) Comparison of direct and iterative artificial neural network forecast approaches in multi-periodic time series forecasting. Expert Syst Appl 36(2): 3839–3844
Hurd HL (2012) A collection of MATLAB programs to do various time series tasks. http://www.stat.unc.edu/faculty/hurd.html . Accessed 12 Feb 2012
Hyndman RJ (2011) Time series data library (TSDL). http://robjhyndman.com/TSDL/
Jose VRR, Winkler RL (2008) Simple robust averages of forecasts: some empirical results. Int J Forecast 24(1): 163–169
Kuncheva LI (2004) Combining pattern classifiers: methods and algorithms. Wiley, NJ
Lemke C, Gabrys B (2010) Meta-learning for time series forecasting and forecast combination. Neurocomputing 73: 2006–2016
Lim CP, Goh WY (2005) The application of an ensemble of boosted elman networks to time series prediction: a benchmark study. J Comput Intell 3(2): 119–126
Makridakis S, Hibon M (2000) The M3 competition: results, conclusions and implications. Int J Forecast 16(4): 451–476
Makridakis S, Andersen A, Carbone R, Fildes R, Hibon M, Lewandowski R, Newton J, Parzen E, Winkler R (1982) The accuracy of extrapolation (time series) methods: results of a forecasting competition. J Forecast 1(2): 111–153
Newbold P, Granger CWJ (1974) Experience with forecasting univariate time series and the combination of forecasts (with discussion). J R Stat Soc A 137(2): 131–165
Reidmiller M, Braun H (1993) A direct adaptive method for faster backpropagation learning: the rprop algorithm. In: IEEE international conference on neural networks (ICNN), San Francisco, USA, pp 586–591
Stock JH, Watson MW (2004) Combination forecasts of output growth in a seven-country data set. J Forecast 23(6): 405–430
Suykens JAK, Vandewalle J (1999) Least squares support vector machines classifiers. Neural Process Lett 9(3): 293–300
Terui N, Van Dijk HK (2002) Combined forecasts from linear and nonlinear time series models. Int J Forecast 18(3): 421–438
Vapnik V (1995) The nature of statistical learning theory. Springer, New York
Winkler RL, Makridakis S (1983a) The combination of forecasts. J R Stat Soc A 146(2): 150–157
Winkler RL, Makridakis S (1983b) Averages of forecasts: some empirical result. Manag Sci 29(9): 987–996
Zhang GP (2003) Time series forecasting using a hybrid ARIMA and neural network model. Neurocomputing 50: 159–175
Zhang G, Patuwo BE, Hu MY (1998) Forecasting with artificial neural networks: the state of the art. Int J Forecast 14(1): 35–62
Zou H, Yang Y (2004) Combining time series models for forecasting. Int J Forecast 20(1): 69–84
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Adhikari, R., Agrawal, R.K. Performance evaluation of weights selection schemes for linear combination of multiple forecasts. Artif Intell Rev 42, 529–548 (2014). https://doi.org/10.1007/s10462-012-9361-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10462-012-9361-z