@misc{girolimetto2023point,
      title={Point and probabilistic forecast reconciliation for general linearly constrained multiple time series},
      author={Daniele Girolimetto and Tommaso {{Di Fonzo}}},
      year={2023},
      eprint={2305.05330},
      archivePrefix={arXiv},
      primaryClass={stat.ME}
}


@article{Wickramasuriya2018OptimalFR,
  title={Optimal Forecast Reconciliation for Hierarchical and Grouped Time Series Through Trace Minimization},
  author={Shanika L. Wickramasuriya and George Athanasopoulos and Rob J Hyndman},
  journal={Journal of the American Statistical Association},
  year={2018},
  volume={114},
  pages={804 - 819}
}

@book{lyche2020numerical,
  title={Numerical linear algebra and matrix factorizations},
  author={Lyche, Tom},
  volume={22},
  year={2020},
  publisher={Springer Nature}
}

@book{meyer2023matrix,
  title={Matrix analysis and applied linear algebra},
  author={Meyer, Carl D and Stewart, Ian},
  year={2023},
  publisher={SIAM}
}

@Manual{rcore,
  title = {R: A Language and Environment for Statistical Computing},
  author = {{R Core Team}},
  organization = {R Foundation for Statistical Computing},
  address = {Vienna, Austria},
  year = {2021},
  url = {https://www.R-project.org/},
}

@article{wilkinson1973symbolic,
 ISSN = {00359254, 14679876},
 URL = {http://www.jstor.org/stable/2346786},
 author = {G. N. Wilkinson and C. E. Rogers},
 journal = {Journal of the Royal Statistical Society. Series C (Applied Statistics)},
 number = {3},
 pages = {392--399},
 publisher = {[Wiley, Royal Statistical Society]},
 title = {Symbolic Description of Factorial Models for Analysis of Variance},
 urldate = {2023-10-16},
 volume = {22},
 year = {1973}
}

@misc{notation,
  title={Notation for forecast reconciliation},
  author={Rob Hyndman and Tommaso {{Di Fonzo}}},
  year={2022},
  url={https://robjhyndman.com/hyndsight/reconciliation-notation.html}
}

@Inbook{Athanasopoulos2020,
author="Athanasopoulos, George
and Gamakumara, Puwasala
and Panagiotelis, Anastasios
and Hyndman, Rob J.
and Affan, Mohamed",
editor="Fuleky, Peter",
title="Hierarchical Forecasting",
bookTitle="Macroeconomic Forecasting in the Era of Big Data: Theory and Practice",
year="2020",
publisher="Springer International Publishing",
address="Cham",
pages="689--719",
abstract="Accurate forecasts of macroeconomic variables are crucial inputs into the decisions of economic agents and policy makers. Exploiting inherent aggregation structures of such variables, we apply forecast reconciliation methods to generate forecasts that are coherent with the aggregation constraints. We generate both point and probabilistic forecasts for the first time in the macroeconomic setting. Using Australian GDP we show that forecast reconciliation not only returns coherent forecasts but also improves the overall forecast accuracy in both point and probabilistic frameworks.",
isbn="978-3-030-31150-6",
doi="10.1007/978-3-030-31150-6_21",
url="https://doi.org/10.1007/978-3-030-31150-6_21"
}


@article{ATHANASOPOULOS201760,
title = {Forecasting with temporal hierarchies},
journal = {European Journal of Operational Research},
volume = {262},
number = {1},
pages = {60-74},
year = {2017},
issn = {0377-2217},
doi = {https://doi.org/10.1016/j.ejor.2017.02.046},
url = {https://www.sciencedirect.com/science/article/pii/S0377221717301911},
author = {George Athanasopoulos and Rob J. Hyndman and Nikolaos Kourentzes and Fotios Petropoulos},
keywords = {Forecasting, Hierarchical forecasting, Temporal aggregation, Reconciliation, Forecast combination},
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.}
}

@article{hyndman2011,
author = {Hyndman, Rob J. and Ahmed, Roman A. and Athanasopoulos, George and Shang, Han Lin},
title = {Optimal Combination Forecasts for Hierarchical Time Series},
year = {2011},
issue_date = {September, 2011},
publisher = {Elsevier Science Publishers B. V.},
address = {NLD},
volume = {55},
number = {9},
issn = {0167-9473},
url = {https://doi.org/10.1016/j.csda.2011.03.006},
doi = {10.1016/j.csda.2011.03.006},
abstract = {In many applications, there are multiple time series that are hierarchically organized and can be aggregated at several different levels in groups based on products, geography or some other features. We call these ''hierarchical time series''. They are commonly forecast using either a ''bottom-up'' or a ''top-down'' method. In this paper we propose a new approach to hierarchical forecasting which provides optimal forecasts that are better than forecasts produced by either a top-down or a bottom-up approach. Our method is based on independently forecasting all series at all levels of the hierarchy and then using a regression model to optimally combine and reconcile these forecasts. The resulting revised forecasts add up appropriately across the hierarchy, are unbiased and have minimum variance amongst all combination forecasts under some simple assumptions. We show in a simulation study that our method performs well compared to the top-down approach and the bottom-up method. We demonstrate our proposed method by forecasting Australian tourism demand where the data are disaggregated by purpose of travel and geographical region.},
journal = {Comput. Stat. Data Anal.},
month = {sep},
pages = {2579–2589},
numpages = {11},
keywords = {Combining forecasts, Hierarchical forecasting, Top-down forecasting, Bottom-up forecasting, Reconciling forecasts, GLS regression}
}