Open Access
June 2024 A hierarchical spline model for correcting and hindcasting temperature data
Theodoros Economou, Catrina Johnson, Elizabeth Dyson
Author Affiliations +
Ann. Appl. Stat. 18(2): 1709-1728 (June 2024). DOI: 10.1214/23-AOAS1855


Weather observations are important for a wide range of applications although they do pose statistical challenges, such as missing values, errors, flawed outliers and poor spatial and temporal coverage to name a few. A Bayesian hierarchical spline framework is presented here to deal with such challenges in temperature time series. Motivated by a real-life problem, the approach uses penalised splines, constructed hierarchically, to pool the data, along with a discrete mixture distribution to deal with outliers and publicly available global reanalysis data sets (climate model data) to integrate physically constrained information. Efficient Bayesian implementation is achieved using conditional conjugacy, which allows thorough model checking and uncertainty quantification. Fitting the model to daily maximum temperature illustrates its flexibility in capturing temporal structures, in pooling of the information and in outlier detection. The model is used to hindcast the time series 50 years into the past while maintaining uncertainty at reasonable levels.

Funding Statement

TE was funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 856612 and the Government of Cyprus.


We thank the Afghanistan Meterological Department for sharing their temperature data and for giving their permission to share a data file to accompany this paper. This work was supported by the Asia Regional Resilience to a Changing Climate (ARRCC) programme, which was funded by the UK Foreign, Commonwealth and Development Office (FCDO). The ERA5 data (Hersbach et al. (2018)) was downloaded from the Copernicus Climate Change Service (C3S) Climate Data Store. The results contain modified Copernicus Climate Change Service information 2020. Neither the European Commission nor ECMWF is responsible for any use that may be made of the Copernicus information or data it contains.


Download Citation

Theodoros Economou. Catrina Johnson. Elizabeth Dyson. "A hierarchical spline model for correcting and hindcasting temperature data." Ann. Appl. Stat. 18 (2) 1709 - 1728, June 2024.


Received: 1 August 2022; Revised: 1 November 2023; Published: June 2024
First available in Project Euclid: 5 April 2024

Digital Object Identifier: 10.1214/23-AOAS1855

Keywords: Bayesian splines , MCMC , mixture model , Outliers , random effects

Rights: Copyright © 2024 Institute of Mathematical Statistics

Vol.18 • No. 2 • June 2024
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