Abstract
An individual seasonal indices (ISI) method and two group seasonal indices (GSI) methods proposed in the literature are compared, based on two models. Rules have been established to choose between these methods and insights are gained on the conditions under which one method outperforms the others. Simulation findings confirm that using the rules improves forecasting accuracy against universal application of these methods.
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Appendices
Appendix A. Rule for the mixed model
The forecast for the ith series, hth season, in year r+1 using ISI is
MSE is
The same forecast using DGSI is
MSE is
MSEISI i > MSEDGSI i if and only if
The forecast of hth season in year r+1 for the ith item using WGSI is
MSE is
MSEISI i > MSEWGSI i if and only if
Appendix B. Rule for the additive model
The additive model is specified as
Forecast for item i, the hth season in year r+1 using ISI is
MSE of the forecast is
The GSI estimator is given as
Therefore, the forecast of item i, the hth season in year r+1 is
MSE of the forecast is as follows:
The difference in MSE for the forecasts using ISI and GSI is:
MSEGSI i < MSEISI i if and only if
Appendix C. Maximum likelihood estimators for the mixed model
where ∑q h=1 S h =q, and ε i,th ∼ iid N(0,σ i 2).
Hence, differentiating with respect to μ i and S h (h=1,…,q) and with respect to σ i 2,
If the q equations ∂logL/∂S h =0 (h=1,…,q) are all satisfied, then:
And so, summing all of these equations:
This gives the same condition as the first equation ∂logL/∂μ i =0
So, if all of the q equations are satisfied, the first equation will also be satisfied. Solving these q equations gives:
Summing, ∑r t=1 ∑q h=1 Y ith =r μ i ∑q h=1 S h
Since the mixed model assumes that the seasonal indices sum to q (the number of seasons):
And the estimators for S h (h=1,…,q) are given by:
And the final condition gives:
We checked that all the criteria for maximum likelihood estimators have been satisfied. The maximum likelihood estimators for the aggregate mixed model and for the additive model can be obtained by the same method.
Appendix D. Bias properties of ISI, DGSI, and WGSI
Expectation of ISI:
Use Taylor series to approximate E(Y i,1h +Y i,2h +⋯+Y i,rh /∑r t=1 ∑q h=1 Y i,th )
Let X 1=Y i,1h +Y i,2h +⋯+Y i,rh ,
Without loss of generality, suppose h=1,
Then for any h,
Therefore,
Expectation of DGSI:
Expectation of WGSI:
Both ISI and WGSI are maximum likelihood estimators, one at the individual level and one at the aggregate level. E(ISI i,h )=S h +(S h −1)σ i 2/qr μ i 2 and, therefore, E(WGSI h )=S h +σ A 2(S h −1)/qr μ A 2.
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Chen, H., Boylan, J. Use of individual and group seasonal indices in subaggregate demand forecasting. J Oper Res Soc 58, 1660–1671 (2007). https://doi.org/10.1057/palgrave.jors.2602310
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DOI: https://doi.org/10.1057/palgrave.jors.2602310