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Addressing Covariate Shift in Industrial Soft Sensing Using Conditional Density Ratio-Based Reweighting
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DOI：10.1109/tii.2026.3687271.png)
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Abstract
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Data-driven soft sensors in industrial processes often suffer from performance degradation due to covariate shift, where process variable distributions drift over time while the input-output relationship remains unchanged. Existing adaptation methods either struggle to capture complex distribution changes or require additional quality measurements that is unavailable in real time. To address these challenges, this paper proposes a covariate shift adaptation framework for industrial soft sensing using a conditional energy-based model (CEBM). The CEBM models the energies of training and testing data using shared neural network parameters, thereby implicitly defining their probability densities. Since the ratio between two densities depends only on the difference of their energies, the conditional density ratio between the training and testing domains can be obtained directly from the learned energy functions. The estimated density ratio is used to reweight training samples, improving the generalization capability of the soft sensor. To efficiently learn the energy function in CEBM, a Conditional Denoising Score Matching (CDSM) strategy is developed. Furthermore, an online learning mechanism with a replay buffer continuously updates the CEBM using streaming test data and historical samples, enabling real-time density ratio estimation and stable predictive performance under evolving covariate shifts. Experiments on simulated and industrial datasets demonstrate that the proposed framework substantially enhances prediction accuracy.
Keywords:
Covariate shift adaptation
density ratio estimation
energy-based models (EBMs)
industrial soft sensing
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