Forecasting with temporal hierarchies

George Athanasopoulos,Rob J Hyndman,Nikolaos Kourentzes,Fotios Petropoulos

doi:10.1016/j.ejor.2017.02.046

George Athanasopoulos, Rob J Hyndman + Show 2 more

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https://doi.org/10.1016/j.ejor.2017.02.046

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Abstract

This paper introduces the concept of Temporal Hierarchies for time series forecasting. A temporal hierarchy can be constructed for any time series by means of non-overlapping temporal aggregation. Predictions constructed at all aggregation levels are combined with the proposed framework to result in temporally reconciled, accurate and robust forecasts. The implied combination mitigates modelling uncertainty, while the reconciled nature of the forecasts results in a unified prediction that supports aligned decisions at different planning horizons: from short-term operational up to long-term strategic planning. The proposed methodology is independent of forecasting models. It can embed high level managerial forecasts that incorporate complex and unstructured information with lower level statistical forecasts. Our results show that forecasting with temporal hierarchies increases accuracy over conventional forecasting, particularly under increased modelling uncertainty. We discuss organisational implications of the temporally reconciled forecasts using a case study of Accident & Emergency departments.

Highlights

Decision making at the operational, tactical and strategic level is at the core of any organisation
The theoretical results on AutoRegressive Integrated Moving Average (ARIMA) processes from these papers are summarised by Rossana and Seater (1995) as being threefold: (a) temporal aggregation contaminates/complicates the dynamics of the underlying ARIMA(p, d, q) process through the moving average component; (b) as the level of aggregation increases the process at the aggregate level is simplified and converges to an IMA(d, d); and (c) aggregation causes loss in the number of observations resulting in a loss in estimation efficiency
To better illustrate the differences between the three proposed scaling methods we show the different matrices for quarterly data: ΛH = diag σA[4], σS[2A] 1, σS[2A] 2, σQ[11], σQ[12], σQ[13], σQ[14] 2, ΛV = diag σ[4], σ[2], σ[2], σ[1], σ[1], σ[1], σ[1] 2, ΛS = diag 4, 2, 2, 1, 1, 1, 1, where the diagonal elements of ΛH correspond to the error variances of the series that make up the quarterly temporal hierarchy in Figure 1, and the diagonal elements of ΛV are the error variances of each aggregation level k ∈ {4, 2, 1}

Summary

Introduction

Decision making at the operational, tactical and strategic level is at the core of any organisation. Long-run strategic forecasts are usually generated considering high level unstructured information from the business environment. These primarily rely on the skill, judgement and experience of senior management, as accurate long-term statistical forecasts that capture the market dynamics can be very challenging to produce (Kolsarici and Vakratsas, 2015). In contrast short-run operational forecasts are usually generated using structured but limited sources of information, such as past sales Wei (1979) was first to study the effect of temporal aggregation on seasonal ARIMA processes His theoretical findings are in line with the summary by Rossana and Seater (1995)

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