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Table 8: Evaluation results of the optimized Theta model.
Yearly Quarterly Monthly Weekly Daily Other All
Forecast
Horizon
6 8 12 12 14 10 -
Method
SES 25.40 21.90 6.44 10.83 19.68 7.23 11.03
sMAPE (%)
HES 36.76 23.20 8.86 10.78 19.42 5.25 14.79
DES 28.76 14.11 6.78 11.40 20.63 6.52 11.02
LRL 39.21 74.97 12.56 43.42 36.32 11.19 23.45
Theta 22.20 13.93 6.30 10.87 19.76 5.91 9.64
Optimized
Theta
22.84 15.55 6.12 11.24 19.53 6.00 9.77
the top performer of the M3 International
Forecasting Competition (Makridakis and Hibon,
2000), the Theta model, were considered. The Theta
model can be described as a more generic
framework, in which the deseasonalized series are
decomposed in two or more Theta lines, each one of
which represents different amount of information.
The next stage constitutes of the extrapolation of the
decomposed lines via various forecasting
techniques. Then the forecasts are combined and the
final point forecasts are calculated. Originally, the
Theta model implementation was suggesting
decomposition into two symmetric Theta lines,
extrapolation with LRL and SES and simple
combination (equal weights). We investigated
further the dynamics of the Theta model, mostly
considering time series of monthly frequency, into
three specific directions:
1. Accurate estimation of seasonal indices
with the use of shrinkage methods against
the classical decomposition method.
2. Decomposition into up to three Theta lines
and alternative combination weights of the
decomposed forecasts into the final model.
3. Optimizations on the smoothing parameter
of the SES method.
The performance exploration of the proposed
modifications took place on the monthly series of
the M3 competition. The empirical results indicate
an overall performance gain of about 2% compared
to the original implementation. The proposed
model’s superiority on monthly data was verified on
a much larger data set containing more than 20,000
time series.
As far as future work is concerned, there are
many possible paths that could be investigated.
Firstly, a link between the weights of the Theta
forecasts into the final model with the forecasting
horizon should be investigated. Secondly, the
selection of the “appropriate” Theta lines should be
explored also as a matter of the qualitative and
quantitative characteristics of each series. Thirdly, a
framework for the selection of the most proper
extrapolation technique for each Theta line is to be
investigated. Finally, the theoretical underpinnings
of the optimized Theta model have to be examined.
REFERENCES
Assimakopoulos, V. Nikolopoulos, N., 2000. The theta
model: a decomposition approach to forecasting,
International Journal of Forecasting, Vol. 16, No. 4,
pp. 521-530.
Crone, S. F., Hibon, M., Nikolopoulos K., 2011. Advances
in forecasting with neural networks? Empirical
evidence from the NN3 competition on time series
prediction, International Journal of Forecasting, Vol.
27, No. 3, pp. 635-660.
James, W., Stein, C., 1961. Estimation with quadratic loss.
In Proceedings of the 4th Berkeley Symposium on
Mathematical Statistics and Probability, Vol. 1.
Berkeley, CA: University of California Press, pp. 361–
379.
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, Journal of Forecasting, Vol. 1, pp. 111–
153.
Makridakis, S., Hibon, M., 2000. The M3-Competition:
Results, conclusions and implications, International
Journal of Forecasting, Vol. 16, No. 4, pp. 451-476.
Miller, D., Williams, D., 2003. Shrinkage estimators of
time series seasonal factors and their effect on
forecasting accuracy. International Journal of
Forecasting, Vol. 19, No. 4, pp. 669-684.
Lemon, G. H., Krutchkoff, R. G., 1969. Smooth empirical
Bayes estimators: with results for the binomial and
normal situations. In Proceedings of the Symposium on
Empirical Bayes Estimation and Computing in
Statistics, Texas Tech University Mathematics Series
No.6, pp. 110–140.
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