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Probabilistic forecast reconciliation: Properties, evaluation and score optimisation.

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Title: Probabilistic forecast reconciliation: Properties, evaluation and score optimisation.
Authors: Panagiotelis, Anastasios1 (AUTHOR) anastasios.panagiotelis@sydney.edu.au, Gamakumara, Puwasala2 (AUTHOR) puwasala.gamakumara@gmail.com, Athanasopoulos, George1,3 (AUTHOR) george.athanasopoulos@monash.edu, Hyndman, Rob J.2 (AUTHOR) rob.hyndman@monash.edu
Source: European Journal of Operational Research. Apr2023, Vol. 306 Issue 2, p693-706. 14p.
Subjects: Time series analysis, Forecasting, Variograms
Abstract: • Forecast reconciliation in the probabilistic setting is rigorously developed. Point forecast reconciliation is extended in a novel way to the probabilistic setting. • Results are derived for the Gaussian and non-Gaussian case. • Theorems on scoring rules are derived with recommendations for forecast evaluation. • A new reconciliation method based on score optimisation and stochastic gradient descent is proposed. • The new methods are shown to improve forecast accuracy in a simulated and empirical setting. We develop a framework for forecasting multivariate data that follow known linear constraints. This is particularly common in forecasting where some variables are aggregates of others, commonly referred to as hierarchical time series, but also arises in other prediction settings. For point forecasting, an increasingly popular technique is reconciliation, whereby forecasts are made for all series (so-called base forecasts) and subsequently adjusted to cohere with the constraints. We extend reconciliation from point forecasting to probabilistic forecasting. A novel definition of reconciliation is developed and used to construct densities and draw samples from a reconciled probabilistic forecast. In the elliptical case, we prove that true predictive distributions can be recovered using reconciliation even when the location and scale of base predictions are chosen arbitrarily. Reconciliation weights are estimated to optimise energy or variogram score. The log score is not considered since it is improper when comparing unreconciled to reconciled forecasts, a result also proved in this paper. Due to randomness in the objective function, optimisation uses stochastic gradient descent. This method improves upon base forecasts in simulated and empirical data, particularly when the base forecasting models are severely misspecified. For milder misspecification, extending popular reconciliation methods for point forecasting results in similar performance to score optimisation. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:• Forecast reconciliation in the probabilistic setting is rigorously developed. Point forecast reconciliation is extended in a novel way to the probabilistic setting. • Results are derived for the Gaussian and non-Gaussian case. • Theorems on scoring rules are derived with recommendations for forecast evaluation. • A new reconciliation method based on score optimisation and stochastic gradient descent is proposed. • The new methods are shown to improve forecast accuracy in a simulated and empirical setting. We develop a framework for forecasting multivariate data that follow known linear constraints. This is particularly common in forecasting where some variables are aggregates of others, commonly referred to as hierarchical time series, but also arises in other prediction settings. For point forecasting, an increasingly popular technique is reconciliation, whereby forecasts are made for all series (so-called base forecasts) and subsequently adjusted to cohere with the constraints. We extend reconciliation from point forecasting to probabilistic forecasting. A novel definition of reconciliation is developed and used to construct densities and draw samples from a reconciled probabilistic forecast. In the elliptical case, we prove that true predictive distributions can be recovered using reconciliation even when the location and scale of base predictions are chosen arbitrarily. Reconciliation weights are estimated to optimise energy or variogram score. The log score is not considered since it is improper when comparing unreconciled to reconciled forecasts, a result also proved in this paper. Due to randomness in the objective function, optimisation uses stochastic gradient descent. This method improves upon base forecasts in simulated and empirical data, particularly when the base forecasting models are severely misspecified. For milder misspecification, extending popular reconciliation methods for point forecasting results in similar performance to score optimisation. [ABSTRACT FROM AUTHOR]
ISSN:03772217
DOI:10.1016/j.ejor.2022.07.040