Forecasting with multivariate temporal aggregation: The case of promotional modelling

By | September 8, 2015

N. Kourentzes and F. Petropoulos, 2015, International Journal of Production Economics, 181: 145-153.

Demand forecasting is central to decision making and operations in organisations. As the volume of forecasts increases, for example due to an increased product customisation that leads to more SKUs being traded, or a reduction in the length of the forecasting cycle, there is a pressing need for reliable automated forecasting. Conventionally, companies rely on a statistical baseline forecast that captures only past demand patterns, which is subsequently adjusted by human experts to incorporate additional information such as promotions. Although there is evidence that such process adds value to forecasting, it is questionable how much it can scale up, due to the human element. Instead, in the literature it has been proposed to enhance the baseline forecasts with external well-structured information, such as the promotional plan of the company, and let experts focus on the less structured information, thus reducing their workload and allowing them to focus where they can add most value. This change in forecasting support systems requires reliable multivariate forecasting models that can be automated, accurate and robust. This paper proposes an extension of the recently proposed Muliple Aggregation Prediction Algorithm (MAPA), which uses temporal aggregation to improve upon the established exponential smoothing family of methods. MAPA is attractive as it has been found to increase both the accuracy and robustness of exponential smoothing. The extended multivariate MAPA is evaluated against established benchmarks in modelling a number of heavily promoted products and is found to perform well in terms of forecast bias and accuracy. Furthermore, we demonstrate that modelling time series using multiple temporal aggregation levels makes the final forecast robust to model misspecification.

Download paper.

Update: R code for MAPAx is now available!

2 thoughts on “Forecasting with multivariate temporal aggregation: The case of promotional modelling

  1. forecaster

    Very nice article. Hyndman ( says that it is not possible to add regressors in ETS. However, in this article I see that you have used regressors in ETS. Has there been some improvements in ETS, such that exogenous variables can now be incorporated ? Would it be possible to add these functions (ETSx/MAPAx) in TStools which by the way is an exceptionally well written package in R which I use at my work frequently?

    1. Nikos Post author

      Thank you, several members of the Lancaster Centre for Forecasting have contributed to TStools and the idea is to keep it updated with our newest research.

      I think we are far from a full understanding of how to best add regressors to ETS and I agree with Rob’s comment. Nonetheless a lot of ETSx models can be easily rewritten in ARIMAX or transfer function equivalent (at least in the way that ETSx is formulated in this paper) in which case we have good understanding how to deal with them. This is particularly relevant to business time series that often the basic forms of ETS (those that have some ARIMA equivalent) work well. Of course this is not true for all ETS formulations. Also keep in mind that in this paper our focus was solely on point forecasts, which makes this an easier problem.

      My view is that there is a more fundamental problem with ETS or ETSx which is related with how we find their parameters, yet the forecasts we are still useful in practice. I hope to have a paper finished soon to back this up. So the theoretical properties of the models may well be violated, yet they still produce relatively good forecasts for typical operational forecasting problems.

      For MAPAx the discussion is more complicated, due to the temporal aggregation. It can be shown that (a simplified) MAPA implies a much more complex cost function for finding the parameters of a functionally equivalent ETS. The interesting thing here is that this cost function overcomes some of the limitations of the conventional way to estimate parameters, which also explains the better performance over ETS, even without the additional benefits of mitigating model uncertainty. Therefore, with regards to the validity of including regressors a similar comments to ETSx can be made, since a functional equivalence can be shown.

      With regards to your question, in TStools there is a very flexible exponential smoothing implementation in the es function. This permits regressors using the xreg input, so you can use it to build ETSx forecasts. To match the model in the paper you will also need to preprocess the promotions with PCA. Currently there is no MAPAx in R, but the plan is to write a MAPA function based on es in the near future, which will allow running MAPAx models. We did the experiments of the paper in Matlab, so we do not have code that can be easily distributed.


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