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Multivariate probabilistic forecasting using Bayesian model averaging and copulas
Annette Möller, Alex Lenkoski, Thordis L. Thorarinsdottir
TL;DR
Univariate postprocessing does not capture the dependence needed for joint weather forecasts, and explicit joint modeling becomes cumbersome for heterogeneous marginals. The paper couples independently estimated BMA marginals with a Gaussian copula; on Pacific Northwest forecasts, the resulting distributions are calibrated and sharp, with improved verification results over independence.
Problem
Many postprocessing methods consider one weather quantity at a fixed lead time and omit potential spatial dependencies, despite the need to preserve multivariate correlation structure.
Source: §1 · p. 2Method
The framework fits BMA predictive marginals independently for each weather variable and joins them using a Gaussian copula.
Source: §1 · p. 2Results
Across the 60 Northwest stations, determinant sharpness and energy score improve for the copula versus independence, Euclidean score is essentially unchanged, and calibration appears improved.
Source: §3.2 · p. 12Takeaways & Limitations
The BMA marginals can be replaced by other marginal-distribution methods without substantially affecting the overall copula methodology.
Source: §4 · p. 13The method assumes a constant correlation factor across observations, which the authors note may be an oversimplification.
Source: §4 · p. 13
Abstract
from arXiv · showhide
We propose a method for post-processing an ensemble of multivariate forecasts in order to obtain a joint predictive distribution of weather. Our method utilizes existing univariate post-processing techniques, in this case ensemble Bayesian model averaging (BMA), to obtain estimated marginal distributions. However, implementing these methods individually offers no information regarding the joint distribution. To correct this, we propose the use of a Gaussian copula, which offers a simple procedure for recovering the dependence that is lost in the estimation of the ensemble BMA marginals. Our method is applied to 48-h forecasts of a set of five weather quantities using the 8-member University of Washington mesoscale ensemble. We show that our method recovers many well-understood dependencies between weather quantities and subsequently improves calibration and sharpness over both the raw ensemble and a method which does not incorporate joint distributional information.
1 Introduction
Univariate postprocessing can miss dependencies needed for joint weather forecasts, while direct joint modeling becomes cumbersome for heterogeneous variables. The paper combines BMA marginals with a Gaussian copula and applies the framework to forecasts of five weather quantities.
1 Introduction
- Ensembles are often biased and underdispersed, while single-quantity postprocessing can ignore spatial dependencies and violate the correlation structure of forecasts and observations.The introduction motivates probabilistic postprocessing that returns full predictive distributions.Source: §1 · pp. 1–2
- In higher dimensions, joint parametric modeling becomes cumbersome when variables have different marginal distributions, motivating separate modeling of marginals and dependence.Copulas model the multivariate dependence structure of rank statistics.Source: §1 · p. 2
- The proposed framework fits BMA marginals independently for each weather variable, then joins them with a Gaussian copula.The authors describe the resulting marginals as calibrated and sharp.Source: §1 · p. 2
- The case study uses 48-hour forecasts of five weather variables from the eight-member University of Washington ensemble at 60 Pacific Northwest locations in 2008.The variables are maximum and minimum temperature, sea level pressure, precipitation accumulation, and maximum wind speed.Source: §1 · p. 2
- Unlike the Schaake Shuffle and ECC, the approach estimates a Gaussian dependence model with one correlation matrix, which the authors suggest may scale better to high dimensions.The alternative methods inherit multivariate ranks from historical observations or the ensemble, respectively.Source: §1 · p. 2
2 Data and methods
The methods combine variable-specific ensemble BMA marginals with a Gaussian copula dependence model, and assess the resulting multivariate forecasts using a 2008 UWME case study and multivariate verification measures.
2.1 Data
- The case study uses daily 48-hour forecasts from the eight-member UWME for 60 observation locations, with additional 2006–2007 data for training and correlation estimation.The study covers calendar year 2008; locations had 95–271 complete days, and forecasts were interpolated from a 12 km grid to observation sites.Source: §2.1 · p. 3
2.2 Ensemble Bayesian model averaging
- Ensemble BMA forms a predictive mixture by associating each ensemble member with a variable-appropriate kernel density.Temperature and pressure use Gaussian kernels, wind speed uses gamma kernels, and precipitation uses a two-part model with a gamma density for cube-root positive amounts.Source: §2.2 · pp. 3–4
2.3 Gaussian Copulas
- A Gaussian copula joins BMA marginals through a correlation matrix estimated from latent Gaussian observations, then samples joint forecasts while retaining those marginals.The case-study framework shares one correlation matrix across timepoints; the authors note this assumption can be relaxed.Source: §2.3 · pp. 5–6
2.4 Multivariate forecast verification
- Multivariate verification assesses calibration with rank histograms and sharpness with determinant sharpness, alongside Euclidean error and energy score.Scores are averaged across locations and test times; energy score and Euclidean error are negatively oriented, so smaller values indicate better performance.Source: §2.4 · pp. 6–7
3 Results
Across the 60 Northwest stations, the Gaussian copula approach improves predictive performance over independence on the energy score and calibration, while the KSEA analysis shows the benefit of modeling dependence beyond the shared marginals. The results also show recognizable temperature and pressure correlations and improved calibration over the raw ensemble.
3 Results; 3.1 Results at Sea-Tac Airport
- The 2008 analysis covers maximum and minimum temperature, sea level pressure, maximum wind speed, and precipitation; BMA univariate postprocessing is applied separately at each observation location, using a 40-day sliding training period for parameter estimation.In the KSEA-focused analysis, all available 2007 data were used to estimate a single correlation matrix, which was carried forward for 2008 verification.Source: §3 · pp. 7–8, §3.1 · p. 8
3.1 Results at Sea-Tac Airport
- At KSEA, estimated correlations are negative between pressure and both temperatures, and positive between minimum and maximum temperature; the joint predictive distribution also shows bias correction and greater spread than the ensemble.The joint distribution is constructed using the estimated correlation matrix and BMA marginals.Source: §3.1 · pp. 8–9
- At KSEA, the copula approach outperforms independence across the reported metrics, with better calibration, greater sharpness, and a lower energy score.Both postprocessed approaches improve on the raw ensemble in all metrics except determinant sharpness.Source: §3.1 · pp. 9–10, Table 3 · p. 10
3.1 Results at Sea-Tac Airport; 3.2 Results over the Northwest US
- At KSEA and across the Northwest stations, both postprocessing approaches improve multivariate calibration over the raw ensemble, while the copula histograms have more-filled highest bins than independence.Neither approach produces a completely uniform rank histogram at KSEA.Source: §3.1 · p. 10, Figure 2 · p. 10, §3.2 · pp. 11–12
3.2 Results over the Northwest US
- Across 60 stations, minimum and maximum temperatures are mostly positively correlated, while minimum temperature and pressure are negatively correlated with a spatial pattern in the estimates.Minimum-temperature/pressure correlations are mostly between −0.2 and −0.4, with stronger negative estimates in Eastern Washington and Oregon.Source: §3.2 · pp. 10–11, Figure 3 · p. 11
- Across 60 Northwest stations, the copula approach improves the energy score and reliability index over independence, with improved determinant sharpness and essentially unchanged Euclidean scores.The verification scores are averaged over all available days in 2008.Source: Table 4 · p. 12, §3.2 · p. 12
4 Conclusions
The Gaussian-copula method combines postprocessed marginal distributions into a joint forecast, yielding calibrated and sharp distributions in the Pacific Northwest case study. The authors emphasize its flexibility, while noting limits in correlation estimation and scope.
4 Conclusions
- The method yielded calibrated and sharp joint distributions in the Pacific Northwest case study, retaining ensemble-BMA marginals with little additional computation.The approach applies a Gaussian copula after forming univariate marginals.Source: §4 · p. 12
- Other methods for forming marginal distributions can replace ensemble BMA without substantially changing the overall copula methodology.BMA was used because it was established for several weather quantities and supported by implementation software.Source: §4 · p. 13
- Unlike Schaake shuffle and ECC, the method uses a Gaussian copula that need not enforce observed rank structures and models dependence after post-processing.The authors suggest this may scale better to high dimensions, but call for further comparison in ensemble post-processing.Source: §4 · p. 13
- The latent-factor and correlation estimation procedure is deliberately simple; more involved strategies, including a Bayesian approach, remain to be assessed for predictive gains.The paper identifies the Bayesian approach of Hoff (2007) as one such strategy.Source: §4 · p. 13
- The Gaussian-copula estimation assumes a constant correlation factor across observations, which the authors acknowledge may oversimplify dependence.Preliminary investigations found essentially no added benefit from time-varying correlations in the case study; reconsideration may be useful in higher dimensions.Source: §4 · p. 13
- The study targets day-to-day weather rather than extremes; spatial multivariable extensions raise dimensionality and spatial-covariation challenges, for which Gaussian Markov random fields may help.The authors note that an alternative copula may be preferable when joint extreme weather is the central concern.Source: §4 · pp. 13–14