VarFedFuse: An optimization-driven Bayesian federated variational fusion framework for uncertainty-aware multi-sensor drought monitoring

Hasnain Iftikhar; Paulo Canas Rodrigues; Osama Abdulaziz Alamri; Basim S. O. Alsaedi; Hasnain Iftikhar; Paulo Canas Rodrigues; Osama Abdulaziz Alamri; Basim S. O. Alsaedi

doi:10.3934/math.20261366

AIMS Mathematics

2026, Volume 11, Issue 10: 35119-35155. doi: 10.3934/math.20261366

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Research article

VarFedFuse: An optimization-driven Bayesian federated variational fusion framework for uncertainty-aware multi-sensor drought monitoring

1.
Department of Statistics, University of Peshawar, Peshawar, Pakistan
2.
Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan
3.
Department of Statistics, Universidade Federal da Bahia, Salvador, Brazil
4.
Department of Business Management, University of Pretoria, Pretoria 0002, South Africa
5.
Department of Statistics, University of Tabuk, Tabuk 71491, Saudi Arabia

Received: 26 July 2026 Revised: 26 September 2026 Accepted: 15 October 2026 Published: 27 October 2026
MSC : 62M10, 62F15, 62G08, 68T07, 68T09

This study proposes VarFedFuse, an optimization-driven Bayesian federated variational fusion framework for privacy-preserving and uncertainty-aware drought monitoring from heterogeneous environmental observations. The framework integrates local Bayesian variational inference, uncertainty-weighted federated posterior aggregation, graph-attention-based adaptive sensor weighting, and transformer-based temporal encoding. Using the law of total variance, the framework further decomposes predictive uncertainty into data uncertainty, uncertainty propagated through the fused latent drought representation, and model-parameter uncertainty, providing non-overlapping, interpretable sources of prediction uncertainty. We evaluated the proposed framework using multi-source environmental observations collected from five drought-prone regions representing North America, Western Europe, the Mediterranean, Central Asia, and Sub-Saharan Africa. We used multi-source observations of the vegetation condition, land-surface temperature, precipitation, temperature, potential evapotranspiration, and soil moisture, along with Standardized Precipitation Evapotranspiration Index (SPEI)-based drought conditions as the prediction target. VarFedFuse was evaluated against centralized deep fusion (CDF), federated averaging (FedAvg), Bayesian committee machine (BCM), uncertainty-aware federated learning (UAFL), and variational federated learning (VFL). VarFedFuse achieved an RMSE of $ 0.48\pm0.03 $, classification accuracy of $ 74.8\pm0.9\% $, and extreme-drought recall of $ 78.2\pm1.7\% $, compared with $ 0.53\pm0.02 $, $ 72.1\pm1.0\% $, and $ 76.3\pm2.0\% $, respectively, for the strongest competing VFL baseline. It also achieved an expected calibration error of $ 0.041\pm0.003 $ and a negative log-likelihood of $ 0.89\pm0.02 $. Predictive uncertainty increased during drought transitions and under lower sensor agreement, while adaptive fusion assigned greater influence to the most informative and reliable sensing modalities as drought evolved. These results demonstrate that VarFedFuse provides a unified framework for accurate drought assessment, calibrated uncertainty quantification, adaptive multi-sensor interpretation, and privacy-preserving distributed environmental-monitoring.
- Bayesian federated variational fusion,
- Bayesian variational inference,
- multi-sensor data fusion,
- uncertainty quantification,
- drought monitoring
Citation: Hasnain Iftikhar, Paulo Canas Rodrigues, Osama Abdulaziz Alamri, Basim S. O. Alsaedi. VarFedFuse: An optimization-driven Bayesian federated variational fusion framework for uncertainty-aware multi-sensor drought monitoring[J]. AIMS Mathematics, 2026, 11(10): 35119-35155. doi: 10.3934/math.20261366

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Abstract

This study proposes VarFedFuse, an optimization-driven Bayesian federated variational fusion framework for privacy-preserving and uncertainty-aware drought monitoring from heterogeneous environmental observations. The framework integrates local Bayesian variational inference, uncertainty-weighted federated posterior aggregation, graph-attention-based adaptive sensor weighting, and transformer-based temporal encoding. Using the law of total variance, the framework further decomposes predictive uncertainty into data uncertainty, uncertainty propagated through the fused latent drought representation, and model-parameter uncertainty, providing non-overlapping, interpretable sources of prediction uncertainty. We evaluated the proposed framework using multi-source environmental observations collected from five drought-prone regions representing North America, Western Europe, the Mediterranean, Central Asia, and Sub-Saharan Africa. We used multi-source observations of the vegetation condition, land-surface temperature, precipitation, temperature, potential evapotranspiration, and soil moisture, along with Standardized Precipitation Evapotranspiration Index (SPEI)-based drought conditions as the prediction target. VarFedFuse was evaluated against centralized deep fusion (CDF), federated averaging (FedAvg), Bayesian committee machine (BCM), uncertainty-aware federated learning (UAFL), and variational federated learning (VFL). VarFedFuse achieved an RMSE of $ 0.48\pm0.03 $, classification accuracy of $ 74.8\pm0.9\% $, and extreme-drought recall of $ 78.2\pm1.7\% $, compared with $ 0.53\pm0.02 $, $ 72.1\pm1.0\% $, and $ 76.3\pm2.0\% $, respectively, for the strongest competing VFL baseline. It also achieved an expected calibration error of $ 0.041\pm0.003 $ and a negative log-likelihood of $ 0.89\pm0.02 $. Predictive uncertainty increased during drought transitions and under lower sensor agreement, while adaptive fusion assigned greater influence to the most informative and reliable sensing modalities as drought evolved. These results demonstrate that VarFedFuse provides a unified framework for accurate drought assessment, calibrated uncertainty quantification, adaptive multi-sensor interpretation, and privacy-preserving distributed environmental-monitoring.

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