Online forecast reconciliation using linear models

2026-06-22

AI Summary

This work addresses the inconsistency among multi-granularity forecasts in online hierarchical time series prediction by proposing a reconciliation method that explicitly models hierarchical relationships through a graph structure. The approach characterizes forecast residuals using a matrix normal distribution and formulates a multivariate linear regression framework, integrating ridge regression, Bayesian estimation, and shrinkage principles. An efficient online recursive inference mechanism is developed to enable adaptive forecast reconciliation and uncertainty quantification. The method is validated on a district heating load forecasting task, demonstrating its effectiveness. To support practical deployment, the authors release PyOnlineForecast, an open-source toolkit for online hierarchical forecasting.

Abstract

We present a framework for online and adaptive forecasting and hierarchical reconciliation using linear regression models. We begin by formalizing hierarchies using graphs, and motivated by their structure, formulate a multivariate linear model using the matrix normal distribution to characterize residuals. Parameter estimation is posed as a ridge regression problem and applied to hierarchical forecast reconciliation. The connections between ridge regression, Bayesian estimation and shrinkage for hierarchical reconciliation are discussed, and results for uncertainty quantification in parameters and forecasts are provided. Based on the ridge regression formulation, a recursive inference scheme inspired by recursive least squares is described. The algorithm is implemented in the PyOnlineForecast package. Finally, the proposed methodology is demonstrated on a case study for district heating load forecasting using a temporal hierarchy. Our results provide a reference for implementation of forecast reconciliation via multivariate linear models in an online setting. The case study furthermore highlights practical considerations of using temporal hierarchies in an online setting and demonstrates the usefulness of the proposed framework and implementation, both for district heating load forecasting and more generally for online hierarchical forecasting.

Problem

• online forecasting
• hierarchical reconciliation
• linear models
• temporal hierarchy
• forecast coordination

Innovation

• online forecast reconciliation
• ridge regression
• matrix normal distribution
• recursive inference
• temporal hierarchy