Modern industrial data is commonly dynamic and contains outliers, which challenges the accurate isolation of faulty variables in abnormal situations. To deal with process dynamics, a robust stacked probabilistic latent variable model is proposed, which is formed by stacking a series of static probabilistic latent variable models. A fault indicator matrix with the Bernoulli-Gaussian prior is constructed to indicate which process variables are faulty. The Bernoulli-Gaussian prior neatly accommodates the stacked structure of the indicator matrix so that rows corresponding to normal variables will shrink to zero. The stacked probabilistic latent variable model is further extended to deal with outliers by introducing an outlier indicator vector with the Beta-Bernoulli prior. The location and magnitude of the outliers can be successfully identified. Based on the robust stacked model, a variational Bayesian inference algorithm is developed to estimate unknown parameters. By using the piecewise affine approximation, the proposed fault isolation method can be extended to deal with nonlinear processes. The effectiveness and superiority of the method are illustrated by application studies to a simulation case and an industrial boiler case. Note to Practitioners —Data-driven fault isolation methods are critical to helping find the accurate root causes of industrial faults. While designing the fault isolation procedures, traditional data-based methods seldom consider autocorrelation and outliers characteristics in the collected process data simultaneously. This paper proposes an accurate and robust fault isolation method based on the stacked probabilistic latent variable model. For the full implementation of the model, it is necessary to: 1) construct the stacked structure of the fault indicator matrix based on the Bernoulli-Gaussian prior; 2) establish the outlier indicator vector with the Beta-Bernoulli prior to determine the location and magnitude of the outliers; 3) optimize the robust stacked model through the variational Bayesian inference algorithm with appropriately selected priors; 4) extract the faulty information of the industrial data to further enhance the faulty isolation performance. The two case studies have shown satisfactory fault-locating accuracy with the proposed model.

This paper develops an incipient fault detection and isolation method using the Wasserstein distance, which measures the difference between the probability distributions of normal and faulty data sets from the aspect of optimal transport. For fault detection, a moving window based approach is introduced, resulting in two monitoring statistics that are constructed based on the Wasserstein distance. From analysis of the limiting distribution under multivariate Gaussian case, it is proved that the difference measured by the Wasserstein distance is more sensitive than conventional quadratic statistics like Hotelling’s $T^2$ and Squared Prediction Error (SPE). For non-Gaussian distributed data, a project robust Wasserstein distance (PRW) model is proposed and the Riemannian block coordinate descent (RBCD) algorithm is applied to estimate the Wasserstein distance, which is fast when the number of sampled data is large. In addition, a fault isolation method is further proposed once the incipiently developing fault is detected. Application studies to a simulation example, a continuous stirred tank reactor (CSTR) process and a real-time boiler water wall over-temperature process demonstrate the effectiveness of the proposed method.

Shapley value is a widely used tool in explainable artificial intelligence (XAI), as it provides a principled way to attribute contributions of input features to model outputs. However, estimation of Shapley value requires capturing conditional dependencies among all feature combinations, which poses significant challenges in complex data environments. In this article, EmSHAP (Energy-based model for Shapley value estimation), an accurate Shapley value estimation method, is proposed to estimate the expectation of Shapley contribution function under the arbitrary subset of features given the rest. By utilizing the ability of energy-based model (EBM) to model complex distributions, EmSHAP provides an effective solution for estimating the required conditional probabilities. To further improve estimation accuracy, a GRU (Gated Recurrent Unit)-coupled partition function estimation method is introduced. The GRU network captures long-term dependencies with a lightweight parameterization and maps input features into a latent space to mitigate the influence of feature ordering. Additionally, a dynamic masking mechanism is incorporated to further enhance the robustness and accuracy by progressively increasing the masking rate. Theoretical analysis on the error bound as well as application to four case studies verified the higher accuracy and better scalability of EmSHAP in contrast to competitive methods.

Graph neural networks (GNNs) have emerged as powerful tools for industrial soft sensing, offering the ability to model complex relationships among process variables. However, existing GNN-based soft sensors suffer from two critical limitations: (i) they lack hierarchical modeling capabilities to reflect the multi-level structure of industrial processes, and (ii) their black-box nature hinders interpretability, making it difficult to evaluate the influence of process variables on key performance indicators. These limitations reduce the reliability of GNNs in safety-critical industrial processes. To address these challenges, this paper proposes a tree structure guided GNN framework that integrates a data-driven adjacency matrix with a tree-based process representation derived from prior knowledge, enabling hierarchical feature aggregation and clearer correlation modeling. Based on this, a tree structure guided graph information bottleneck (GIB) method is developed to extract critical subtrees that preserve predictive information while suppressing task-irrelevant redundancies, thereby enhancing interpretability. Furthermore, an intrinsic counterfactual explanation module is introduced to generate actionable anomaly regulation suggestions by identifying minimal and interpretable process adjustments needed to restore key performance indicators to safe operating ranges. Experimental results on simulation and real-world industrial datasets demonstrate that the proposed framework achieves superior predictive accuracy, interpretability, and practical effectiveness in anomaly regulation.

This article develops a novel process monitoring method based on the Wasserstein distance for incipient fault detection. The core idea is to measure the difference between the normal data and the faulty data. For Gaussian distributed process variables, the paper proved that the difference measured by the Wasserstein distance is more sensitive than the Hotelling¡ ⁻ s $T^{2}$ and the Squared Prediction Error (SPE) in the Principal Component Analysis (PCA) framework. For non-Gaussian distributed data, a Project Robust Wasserstein distance (PRW) model under the PCA framework is proposed and an algorithm called Riemannian Block Coordinate Descent (RBCD) algorithm is used to solve this model, which is fast when the number of sampled data is large. An application study to a glass melter demonstrate the effectiveness of the proposed method.

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.