Industrial data often consist of continuous variables (CVs) and binary variables (BVs), both of which provide crucial information about process operating conditions. Due to the coupling between industrial systems or equipment, these hybrid variables are usually high-dimensional and highly correlated. However, existing methods generally model hybrid variables directly in the observation space and assume independence between the variables to overcome the curse of dimensionality. Thus, they are ineffective at capturing dependencies among hybrid variables, and the effectiveness of process monitoring will be compromised. To overcome the limitations, this study proposes to seek a unified subspace for hybrid variables using the probabilistic latent variable (LV) model. By introducing a low-dimensional continuous LV, the proposed method can avoid the curse of dimensionality while capturing the dependencies between hybrid variables. Nevertheless, the inference of LV is analytically intractable and thus time-consuming due to the heterogeneity of CVs and BVs. To accelerate offline learning and online inference procedures, this study originally derives an analytical Gaussian distribution to approximate the true posterior distribution of the LV, based on which an efficient expectation-maximization algorithm is developed for parameter estimation. The Gaussian approximation is simultaneously optimized with the latest parameters to achieve a high approximation accuracy. The LV is then estimated by the posterior mean of the Gaussian approximation. By mapping the heterogeneous variables into a unified subspace, the proposed method defines three monitoring statistics, which are physically interpretable and thoroughly evaluate the probability of hybrid variables being normal. The effectiveness of the proposed method in detecting anomalies in CVs and BVs is shown through a numerically simulated case and a real industrial case.

Process variables may exhibit both temporal trends and periodic responses, with their fault propagation pathways manifesting in time-domain and frequency-domain causalities, respectively. However, the differing causal perspectives of time-domain and frequency-domain methods can lead to distinct causalities, posing the causal heterogeneity challenge for root cause diagnosis (RCD). Thereupon, we reveal the mechanism of source consistency in Granger causality (GC), that is, the root cause variable provides the most significant predictive information in both time and frequency domains. Accordingly, we propose a causal source consistency analytics (CSCA) framework that achieves time-frequency synergy. First, we design a nonlinear enhancement module to extract temporal features for causal inference. Second, to extract time-domain and frequency-domain GC, we develop a parallel causality learning module, where a differentiable frequency-domain expansion operator is designed along with a temporal prediction submodule. Meanwhile, a time-frequency entropy constraint is constructed to ensure causal significance by inducing sparsity. Finally, a root cause alignment module is proposed to ensure source consistency. A predictive information quantification algorithm, formulated as an eigenvalue decomposition problem, is designed to locate the root cause. We develop an approximate exponential transformation to convert the eigenvalue decomposition into a differentiable source alignment loss. Thus, source consistency can be ensured during end-to-end inference. The validity of CSCA is illustrated through the Tennessee Eastman process and a gas turbine application. CSCA identified the root causes in both examples correctly. Furthermore, ablation studies validate that CSCA enables the time-domain and frequency-domain models to identify consistent root causes, thereby overcoming causal heterogeneity.

Cross-time spatial dependence (i.e., the interaction between different variables at different time points) is indispensable for detecting anomalies in multivariate time series, as certain anomalies may have time delays in their propagation from one variable to another. However, accurately capturing cross-time spatial dependence remains a challenge. Specifically, real-world time series usually exhibits complex and incomprehensible evolutions that may be compounded by multiple temporal states (i.e., temporal patterns, such as rising, fluctuating, and peak). These temporal states mix and overlap with each other and exhibit dynamic and heterogeneous evolution laws in different time series, making the cross-time spatial dependence extremely intricate and mutable. Therefore, a cross-time spatial graph network with fuzzy embedding is proposed to disentangle latent and mixing temporal states and exploit it to meticulously learn cross-time spatial dependence. First, considering that temporal states are diversiform and their mixing modes are unknown, we introduce a fuzzy state set to uniformly characterize potential temporal states and adaptively generate corresponding membership degrees to depict how these states mix. Further, we propose a cross-time spatial graph, quantifying similarities among fuzzy states and sensing their dynamic evolutions, to flexibly learn mutable cross-time spatial dependence. Finally, we design state diversity and temporal proximity constraints to ensure the differences among fuzzy states and the evolution continuity of fuzzy states. Experiments on real-world datasets show that the proposed model outperforms the state-of-the-art models.

Alarm flood classification and alarm prediction are significant ways to assist the on-site operators to manage alarm floods and maintain process safety. The two tasks are interdependent considering the decisive role of different alarm floods on the arising alarms. To give comprehensive consideration of both, this work proposed a hierarchical strategy for alarm flood classification and alarm prediction leveraging the structure feature of alarm floods. The structure feature aims at revealing the sparse causal dependencies among alarm variables. It is achieved by a deep learning model under the guidance of a designed objective function that probabilistically parametrizes causal dependencies with sparsity constraint. Due to its interpretable physical meaning, desirable robustness and discriminate properties are achieved, allowing a win-win situation for both tasks. Based on the structure features, the hierarchical strategy is given, where an overall classifier is built while prediction models are trained for each category. The classifier trained by structure features is predisposed to generate satisfactory early classification results, enabling timely prediction. For the prediction, the structure features are also used to incorporate temporal features to achieve better performance. Experimental results illustrate the interpretability of structure features and show the feasibility of the proposed hierarchical strategy.Note to Practitioners—During alarm floods, the alarm patterns are different than usual and the generic prediction model may fail. The focus of this study is to achieve a win-win situation for both alarm flood classification and prediction, thereby providing comprehensive information required for handling alarm floods. Considering the arising alarm is strongly affected by the type of current alarm flood, a hierarchical alarm flood classification and alarm prediction strategy is given. It exploits the essential characteristic of alarm interactions in alarm floods to generate robust and early classification results, allowing timely predictions by category. In this way, the performance of both tasks can be guaranteed. The proposed method requires labeled historical alarm flood data.

Temporal data contain a wealth of valuable information, playing an essential role in various machine-learning tasks. Slow feature analysis (SFA), one of the most classic temporal feature extraction models, has been deeply explored in two decades of development. SFA extracts slowly varying features as high-level representations of temporal data. Its core idea of "slow" has been proven to be consistent with the nature of biological vision and beneficial in capturing significant temporal information for various tasks. So far, SFA has evolved into numerous improved versions and is widely applied in many fields such as computer vision, industrial control, remote sensing, signal processing, and computational biology. However, there currently lacks an insightful review of SFA. In this article, a comprehensive overview of SFA and its extensions is provided for the first time. The formulation and optimization of SFA are introduced. Two mainstream solutions, geometric interpretation, and a gradient-based training method of SFA are presented and discussed. Following that, a taxonomy of the current progress of SFA is proposed. We classify improved versions of SFA into six categories, including dual-input SFA (DISFA), online slow feature analysis (OSFA), probabilistic SFA (PSFA), multimode SFA, nonlinear SFA, and discrete labeled SFA. For each category, we illustrate its main ideas, mathematical principles, and applicable scenarios. In addition, the practical applications of SFA are summarized and presented. Finally, we bring new insights into SFA according to its research status and provide potential research directions, which may serve as a good reference for promoting future work.

Typically, industrial processes possess both temporal and spatial dependencies due to intravariable dynamics and intervariable couplings. The two dependencies have different manifestations, indicating diverse process characteristics. However, the existing methods fail to separate temporal and spatial information well, leading to inappropriate representation and inaccurate fault detection and isolation results. This study proposes an explicit representation and customized fault isolation framework to tackle temporal and spatial characteristics, so as to identify and locate anomalies affecting different dependencies. First, we design a double-level separation method for temporal and spatial information. In the first level, we construct two independent auto-encoding modules to extract temporal correlation and spatial graph structure in parallel. In the second level, we propose an information aliasing loss function to guild the two modules to distinguish between temporal and spatial characteristics, further facilitating information separation. By monitoring the explicit temporal and spatial statistics obtained by the two modules, spatiotemporal dependencies of anomalies can be determined for subsequent isolation. Furthermore, we propose a customized isolation strategy for anomalies in temporal and spatial characteristics. By quantifying changes in intravariable temporal dynamics and intervariable spatial graph structure individually, temporal impact and spatial propagation of faults can be finely characterized and isolated. Three examples are adopted to verify the performance of the proposed framework, including a numerical example, a real condensing system of the thermal power plant process, and the Tennessee Eastman benchmark process.

Industrial process data are usually high-dimensional with dynamic characteristics, and a mix of continuous and binary quantities. However, current dynamic latent variable (DLV) methods primarily focus on analyzing continuous variables (CVs), overlooking the prevalence and significance of binary variables (BVs). BVs often serve as control references, indicating operating conditions or specific states and influencing the behavior of CVs. Integrating BVs into DLV models is crucial for elucidating the correspondence between CVs and BVs and uncovering the real operating patterns of the system. The main challenge lies in effectively accommodating the statistical heterogeneity exhibited by CVs and BVs, while comprehensively investigating their contemporaneous and temporal dependencies. To address this challenge, this study proposes a novel DLV model called hybrid probabilistic slow feature analysis (HPSFA). The HPSFA algorithm is specifically designed to extract slow features (SFs) from CVs while incorporating supervision from BVs. To efficiently infer posterior distributions of SFs, a variational recursive filter (VRF) is developed using the local approximation method, providing closed-form posterior estimations. Leveraging the VRF, an efficient expectation-maximization algorithm is proposed for parameter estimation. For process monitoring, three statistics are designed based on prediction or reconstruction errors, which are separated from dynamic variations and exhibit reduced variability. This reduction in variability enables the definition of narrower control regions while maintaining the desired confidence level. The HPSFA method is thoroughly evaluated through both simulated and real industrial case studies to demonstrate its validity and superior performance over existing approaches. The experimental results show that HPSFA timely detects both static and dynamic anomalies of the hybrid variables, and achieves the highest-fault detection rate (85.89%) while maintaining a considerably low-false alarm rate (2.67%) in the practical industrial case.