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.

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.

To address the challenges encountered by dictionary learning-based monitoring, this article presents a novel pattern discovery scheme for detection and isolation of incipient faults that involves structured sparse coding and sequential dictionary augmentations. Through learning a basic dictionary for normal pattern and augmenting the low-dimensional sparse dictionaries for analyzing different fault patterns, the process signals can be decomposed into fault-free and fault-related components. To guarantee the in-statistical-control status of the fault-free part and improve detection sensitivity, a$\ell _{2}$-penalty is imposed on the sum of coefficient vectors to ensure that the monitoring statistic related to the fault-free part will not exceed the control limit. In addition, two Frobenius norm penalties are imposed on the zero centered coefficient matrix and atom matrix to improve the robustness of signal decomposition. Instead of imposing$\ell _{1}$-sparsity constraint on the atoms, a hard sparsity constraint is used to correctly select fault-related feature variables, so that fault patterns can be better revealed. The informative dictionaries are then incorporated into the moving window-based monitoring strategy, yielding a fault detection and isolation scheme suitable for incipient faults. The superior performance of our proposed approach is validated by application studies involving a numerical example and two practical industrial processes.

Due to operating condition drift, environmental changes, and system oscillations, industrial processes often exhibit nonstationary characteristics that involve both stable long-term trend and fluctuant short-term dynamics. In this article, a novel multiscale gated structure model (MGSM) is proposed for nonstationary process soft sensing, which includes long-term memory chain (stable and low frequency) and short-term dynamic chain (respond to fluctuations). The information decomposed from input data is introduced into the MGSM to learn long-term dependency relationships and dynamic behavior in the nonstationary process. In addition, a novel two-dimensional random missing function is designed to handle randomly missing data, which fully considers the data missing in variable-wise and time-wise dimensions. The proposed model is further constructed for the soft sensing of nonstationary processes with random missing data. Finally, application studies to the Tennessee Eastman process and a thermal power generating process show that the proposed method has significant advantages in the quality prediction of nonstationary process.

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.

To improve the monitoring performance of dictionary learning-based methods, this article proposes a row-column overcomplete structured dictionary learning (RCOSDL) method for fault detection and isolation of industrial processes. Unlike conventional dictionary learning approaches which are overcomplete column-wise, the proposed method involves a dictionary that is overcomplete both row- and column-wise. The introduction of row-column overcomplete dictionary results in monitoring statistics that are more sensitive to incipient faults. In order to incorporate structured information, two graph Laplacian regularization terms, namely, manifold graph Laplacian and full graph Laplacian are considered. While the inherent local geometric structure in the data is preserved in the sparse representation by the manifold graph Laplacian term, the correlation structure between process variables is preserved by using the full graph Laplacian term. Hence, violation in the geometric or correlation structure will be promptly detected. To pinpoint faulty variables, a fault isolation method is developed by imposing the$l_{1}$/$l_{2,1}$-norm constraint on the sparse coefficients. In addition, theoretical property involving the condition for guaranteed fault isolation is presented in Theorem 1. The contributions of this article include the introduction of monitoring statistics based on RCOSDL that are suitable for incipient faults, a new fault isolation scheme as well as theoretical analysis on the condition of guaranteed fault isolation. The better performance of the proposed method is illustrated by applications to numerical studies and practical industrial cases.

This article develops a novel probabilistic monitoring framework for industrial processes with multiple operational conditions. The proposed method is based on the probabilistic linear discriminant analysis (PLDA), which relies on two sets of latent variables, i.e., the between-class and within-class latent variables. In order to deal with the large within-class variations in multi-mode industrial processes, this approach modifies the original PLDA by introducing a separate within-class loading matrix for each operational mode and designs an expectation maximization (EM) algorithm to estimate the model parameters from the training samples. Mode identification for test samples is achieved by investigating the cosine similarity in the between-class latent variables and two monitoring statistics corresponding to within-class latent variables and the residuals are considered for fault detection. To diagnose the process fault, this article further develops a sparse probabilistic generative model based on PLDA for fault isolation. The enhanced performance of the proposed method is illustrated by applications to numerical examples and industrial processes.

In order to improve the performance of fault isolation and diagnosis of principal component analysis (PCA) based methods, this article proposes a novel fault detection and isolation approach using the structured joint sparse PCA (SJSPCA). The objective function involves two regularization terms: the$l_{2,1}$norm and the graph Laplacian. By imposing the$l_{2,1}$norm, SJSPCA is able to achieve row-wise sparsity, and introducing the graph Laplacian term can incorporate structured variable correlation information. The row-sparsity property of$l_{2,1}$norm ensures that the score indices associated with normal variables approaching zero and the graph Laplacian constraint helps the isolation of correlated faulty variables. Once a fault is detected, a two-stage fault-isolation strategy is considered and a score index is calculated for each variable. It is proved that the proposed two-stage strategy is capable of isolating faulty variables. The improved fault-isolation performance of SJSPCA is illustrated by a simulation example and a gas flow fault observed in an industrial blast furnace iron-making process.