A quality related fault detection method and system

Abstract

The present application relates to the technical field of fault detection, and more particularly to a quality-related fault detection method and system. A robust probabilistic slow feature analysis model is constructed, it is assumed that process noise and measurement noise conform to Laplace distribution, a Gaussian scale mixture model is used to re-parameterize it into a mixed form of Gaussian distribution and exponential distribution, dynamic slow feature equations, process variable observation equations and quality variable observation equations are established, and based on historical process variable data and quality variable data after normalization at different times, the parameters of the robust probabilistic slow feature analysis model are determined; based on the statistics of the quality-related slow feature subspace, the quality-related residual subspace and the quality-independent residual subspace decomposed from the model, accurate discrimination of quality-related faults, quality-independent faults and no faults is realized.

Description

Technical Field

[0001] This invention relates to the field of fault detection technology, and in particular to a method and system for detecting quality-related faults. Background Technology

[0002] Fault detection and diagnosis (FDD) is a key technology for ensuring the safe operation of large-scale complex industrial processes, preventing potential equipment damage, and maintaining stable production quality. With the widespread application of sensors and distributed control systems, data-driven FDD methods have become mainstream, especially suitable for complex systems where it is difficult to establish accurate mechanistic models.

[0003] Multivariate statistical process monitoring (MSPM) methods, such as principal component analysis (PCA) and partial least squares (PLS), are widely used. PCA extracts latent variables by maximizing variance, while PLS utilizes the correlation between process variables and quality variables to achieve quality-oriented fault detection. However, PCA is constrained by linear modeling assumptions, which assume that the underlying data follows a Gaussian distribution, variables are linearly correlated, and the process is statically stationary. Furthermore, it lacks quality-related information, making it difficult to accurately identify quality-related faults. PLS suffers from uncertainties in solving optimization problems and is susceptible to contamination by non-quality-related interference. Moreover, both traditional algorithms are primarily static modeling frameworks, failing to effectively characterize the dynamic operating characteristics of industrial processes.

[0004] Current mainstream improved algorithms include Slow Feature Analysis (SFA), Canonical Variate Analysis (CVA), kernel function mapping methods, deep learning algorithms, and subspace decomposition strategies such as Total Partial Least Squares (T-PLS) and Multi-kernel Partial Least Squares (MKPLS). However, most of these methods assume that the data follows a Gaussian distribution and use mean squared error as the loss criterion to build the model. They are weakly resistant to process outliers and non-Gaussian noise. Sensor drift, impulse interference, and environmental noise in industrial settings are typical non-Gaussian heavy-tailed noise. Mean squared error will amplify the weight of outliers, leading to model parameter shifts. To characterize model and data uncertainty, probabilistic graphical models and mixture models have been introduced. Studies have shown that the heavy-tailed characteristics of the Laplace or t-distribution are better at suppressing outlier effects and reducing outlier interference. However, when introducing a heavy-tailed distribution for robust probabilistic modeling, the original orthogonal mathematical structure is disrupted. The temporal smoothing constraints required for slow feature extraction and the correlation constraints required for decoupling the quality subspace interact and cancel each other out, making it impossible for the model to simultaneously achieve accurate decoupling of slow feature extraction and quality-related / irrelevant subspaces under non-Gaussian noise conditions. This creates an inherent contradiction among the three. This contradiction directly results in existing models having to prioritize only one aspect of modeling performance: if robustness under non-Gaussian noise is pursued, weak slow dynamic features of the process will be lost and subspace boundaries will overlap; if the slow feature extraction capability is retained, the model cannot resist heavy-tailed noise interference; if the quality subspace decoupling is strictly achieved, the ability to characterize temporal dynamics will be sacrificed. Ultimately, this leads to frequent false alarms, missed early gradual faults, and inaccurate fault attribution in complex industrial conditions. Summary of the Invention

[0005] Therefore, the technical problem to be solved by the present invention is to overcome the shortcomings of existing quality-related fault detection methods, which cannot simultaneously take into account robust probability modeling, slow feature extraction and accurate decoupling of quality-related and irrelevant subspaces under non-Gaussian noise, resulting in poor fault detection accuracy.

[0006] To address the aforementioned technical problems, this invention provides a method for detecting quality-related faults, comprising: Assuming that both process noise and measurement noise follow a Laplace distribution, a Gaussian scale mixture model is used to reparameterize the Laplace distribution into a form that includes both Gaussian and exponential distributions. Dynamic slow characteristic equations, observation equations for each process variable, and observation equations for each mass variable are established, and a robust probabilistic slow characteristic analysis model is constructed. Based on the normalized historical process variable data and quality variable data at different times, the parameters of the robust probabilistic slow feature analysis model are determined. The normalized actual process variable data and quality variable data are used to obtain statistics for the quality-related slow feature subspace, quality-related residual subspace, and quality-independent residual subspace through a robust probabilistic slow feature analysis model. Based on the statistics of the quality-related slow feature subspace, the quality-related residual subspace, and the quality-independent residual subspace, the system can distinguish between quality-related faults, quality-independent faults, and no faults.

[0007] Preferably, after reparameterizing the Laplace distribution to include both Gaussian and exponential distributions, the process noise and measurement noise obey the following distribution constraints: , , in, For the first Process noise corresponding to each process variable For the first Measurement noise corresponding to each mass variable For process variable indexing, For indexing quality variables, It follows a Gaussian distribution. for Time of the first Gaussian-scaled mixture variables corresponding to each process variable It follows an exponential distribution. The first Laplace scaling parameter, for Time of the first Gaussian-scaled mixture variables corresponding to each quality variable. This is the second Laplacian scaling parameter. Indicates the time.

[0008] Preferably, the dynamic slow characteristic equation is: , The observation equation for each process variable is: , The observation equation for each mass variable is: , in, express Time-quality related slow features For the dimensions of slow features related to quality, It is a diagonal weight matrix. For the first A process load vector For the first A mass load vector, Indicates transpose. express Slow characteristics related to time and quality. for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, for Slow-moving noise characteristics for Time of the first Process noise corresponding to each process variable for Time of the first Measurement noise corresponding to each mass variable For process variable indexing, For indexing quality variables, Indicates the time.

[0009] Preferably, the method for determining the parameters of the robust probabilistic slow feature analysis model based on normalized historical process variable data and quality variable data at different times includes: Based on the normalized historical process variable data and quality variable data at different times, the parameters of the robust probability slow feature analysis model are solved with the goal of maximizing the joint probability likelihood function of the robust probability slow feature analysis model.

[0010] Preferably, the method for optimizing the objective and determining the final optimization objective includes: The joint probability likelihood function of the robust probability slow feature analysis model is transformed into logarithmic form to obtain the logarithmic joint probability likelihood function, with maximizing the logarithmic joint probability likelihood function as the final optimization objective. The log-joint probability likelihood function is: , in, Let be the log-joint probability likelihood function. Let be the probability density function. The parameter matrix to be solved for the robust probabilistic slow feature analysis model is... For historical process variable data after normalization at different times, The data consists of normalized historical quality variables at different points in time. The slow characteristics related to quality at different times. The process load matrix is ​​composed of all process load vectors. The mass load matrix is ​​composed of all mass load vectors. It is a diagonal weight matrix. The first Laplace scaling parameter, This is the second Laplacian scaling parameter. The total number of moments. The total number of process variables. For the total number of quality variables, for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, express Time-quality related slow features for Time of the first Gaussian-scaled mixture variables corresponding to each process variable for Time of the first Gaussian-scaled mixture variables corresponding to each quality variable. , , It follows an exponential distribution. Indicates time, For process variable indexing, Index for quality variables.

[0011] Preferably, the method for solving the parameters of the robust probabilistic slow feature analysis model includes: S21: Initialize the parameter matrix to be solved for the robust probabilistic slow feature analysis model, initialize the Gaussian scale mixture variables corresponding to each process variable and quality variable at each time step, and set the current iteration number. and convergence threshold; S22: Based on the first The parameter matrix to be solved for the nth iteration, the process noise, measurement noise and corresponding Gaussian-scaled mixture variables at each time step, are obtained by Kalman filtering. The posterior distribution of the quality-related slow features at each time step of the next iteration is calculated, and the expected value of the log joint probability likelihood function is calculated. S23: Obtain the first... through variational Bayesian inference. The expected value of the Gaussian-scaled mixture variable corresponding to each process variable and quality variable at each time point of each iteration; S24: Combine the normalized historical process variable data and the quality variable data at each time point, and... Substituting the expected value of the logarithmic joint probability likelihood function, the expected value of the Gaussian-scaled mixture variable corresponding to each process variable and quality variable at each time step, and the posterior distribution of the quality-related slow features at each time step into the joint probability likelihood function, we can solve for the parameter matrix to be solved, thus obtaining the... The parameter matrix to be solved in the next iteration; S25: Determine the... The parameter matrix to be solved in the nth iteration and the nth iteration If the matrix norm of the difference between the parameter matrices to be solved in the next iteration is greater than or equal to the convergence threshold, then let... Return to step S22; otherwise, proceed to step S23. The parameter matrix to be solved in the next iteration is used as the parameter of the robust probabilistic slow feature analysis model.

[0012] Preferably, the first is obtained through variational Bayesian inference. The formula for calculating the expected value of the Gaussian-scaled mixture variable corresponding to each process variable and the mass variable at each time step of the iteration is as follows: , , in, For the first iteration At this moment The expected value of the Gaussian-scaled mixture variables corresponding to each process variable. For the first iteration At this moment The expected value of the Gaussian-scaled mixture variable corresponding to each quality variable. Expressing expectations, , , For the first The first Laplacian scaling parameter of the next iteration. This is the second Laplacian scaling parameter. for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, Indicates the first iteration Time-quality related slow features For the first A process load vector For the first A mass load vector, Indicates transpose. Indicates time, For process variable indexing, Index for quality variables.

[0013] Preferably, the parameter matrix to be solved is solved to obtain the first... The parameter matrix to be solved in the nth iteration, the nth The process of solving the parameter matrix in the next iteration includes: Based on the normalized historical process variable data at each time point, the first The posterior distribution of the quality-related slow features at each time step in the iteration, and the expected value of the Gaussian-scaled mixture variable corresponding to each process variable at each time step, are used to calculate the... The process load matrix within the parameter matrix to be solved in the next iteration; Based on the normalized historical quality variable data at each time point, the first The posterior distribution of the quality-related slow features at each time step in the iteration, and the expected value of the Gaussian-scaled mixture variable corresponding to each quality variable at each time step, are used to calculate the... The mass load matrix within the parameter matrix to be solved in the next iteration; Given a diagonal weight matrix where all diagonal elements are greater than or equal to 0 and less than 1, solve for the... The diagonal weight matrix within the parameter matrix to be solved in the next iteration; By examining the first The expected values ​​of the Gaussian-scaled mixture variables corresponding to the full-time, full-dimensional process variables in the nth iteration are globally averaged to obtain the nth... The first Laplace scale parameter within the parameter matrix to be solved in the next iteration; By examining the first The expected values ​​of the Gaussian-scaled mixture variables corresponding to the full-time series and full-dimensional quality variables in the nth iteration are globally averaged to obtain the nth... The second Laplace scale parameter within the parameter matrix to be solved in the next iteration.

[0014] Preferably, solving the first The methods for solving the diagonal weight matrix within the parameter matrix in the next iteration include: By solving the constructed nonlinear equations, the first... The values ​​of each diagonal element in the diagonal weight matrix within the parameter matrix to be solved in the next iteration; the nonlinear equation is: , in, For the first In the diagonal weight matrix of the nth iteration The values ​​of the diagonal elements, Indicates to Find the partial derivative. , For the dimensions of slow features related to quality, The total number of moments. The total number of process variables. For the total number of quality variables, Indicates the first iteration Time-quality related slow features Indicates the first iteration Time-quality related slow features Indicates the time.

[0015] The present invention also provides a quality-related fault detection system, comprising: The model building module is used to set both process noise and measurement noise to follow a Laplace distribution. Using a Gaussian scale mixture model, the Laplace distribution is reparameterized into a form that includes both Gaussian and exponential distributions. The dynamic slow feature equation, the observation equations for each process variable and the observation equations for each mass variable are established to build a robust probabilistic slow feature analysis model. The parameter determination module is used to determine the parameters of the robust probabilistic slow feature analysis model based on the normalized historical process variable data and quality variable data at different times. The statistics module is used to obtain statistics for the quality-related slow feature subspace, quality-related residual subspace, and quality-independent residual subspace by using a robust probabilistic slow feature analysis model to analyze the normalized actual process variable data and quality variable data. The identification module is used to distinguish between quality-related faults, quality-independent faults, and no faults based on statistics of the quality-related slow feature subspace, the quality-related residual subspace, and the quality-independent residual subspace.

[0016] Compared with the prior art, the above-described technical solution of the present invention has the following advantages:

[0017] This invention discloses a quality-related fault detection method and system, which proposes a robust probabilistic slow feature analysis model. By assuming that process noise and sensor measurement noise in industrial production scenarios both follow a Laplace distribution, and combining this with a Gaussian scale mixture model to complete noise reparameterization reconstruction, the model fully utilizes the strong heavy-tailed distribution characteristics of the Laplace distribution. It abandons the Gaussian distribution assumption and mean squared error loss criterion commonly used in traditional modeling, and optimizes the noise fitting capability from the underlying model architecture. This effectively weakens the adverse effects of sensor signal drift, instantaneous pulse interference, environmental noise, and outlier samples from production anomalies commonly found in industrial settings. It significantly enhances the overall model's ability to resist and suppress non-Gaussian noise and abrupt outliers, fully leverages the heavy-tailed characteristics of the Laplace distribution, and significantly improves the model's ability to resist non-Gaussian noise and outliers in industrial data. This enables more accurate capture of process uncertainties under complex working conditions and improves the accuracy of quality-related fault detection.

[0018] This invention also proposes a hybrid parameter estimation strategy that integrates the expectation-maximization algorithm, variational Bayesian inference, and Kalman filtering and smoothing. Kalman filtering and temporal smoothing are introduced into the E-step of the expectation-maximization algorithm. The posterior distribution of the latent state of slow features is solved based on the temporal recursive relationship of Kalman filtering. Then, variational Bayesian inference is used to update the expected values ​​of the Gaussian-scale mixture variables corresponding to each process variable and quality variable at each time step. In the M-step, independent optimization and update formulas are designed for different components of the parameter matrix to be solved. This hybrid estimation strategy is fully compatible with the modeling system of this invention and can stably complete the estimation of all unknown parameters. It not only estimates the unknown parameters of the model efficiently and accurately but also effectively extracts quality-related slow features while capturing the dynamic characteristics of the process. It solves the problems of static modeling and unstable parameter estimation in traditional methods, providing reliable model support for fault detection. Attached Figure Description

[0019] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings, wherein:

[0020] Figure 1 This is a flowchart illustrating a quality-related fault detection method according to the present invention.

[0021] Figure 2 This is a graph showing the prediction performance of the present invention and other algorithms under IDV13 fault conditions.

[0022] Figure 3 This is a graph showing the monitoring performance of the RRCR algorithm under IDV13 fault conditions.

[0023] Figure 4 This is a graph showing the monitoring performance of the RLVR algorithm under IDV13 fault conditions.

[0024] Figure 5 This is a graph showing the monitoring performance of the MPPLS algorithm under IDV13 fault conditions.

[0025] Figure 6 This is a monitoring performance diagram of the present invention under IDV13 fault conditions.

[0026] Figure 7 This is a graph showing the monitoring performance of the RRCR algorithm under IDV4 fault conditions.

[0027] Figure 8 This is a graph showing the monitoring performance of the RLVR algorithm under IDV4 fault conditions.

[0028] Figure 9 This is a graph showing the monitoring performance of the MPPLS algorithm under IDV4 fault conditions.

[0029] Figure 10 This is a graph showing the monitoring performance of the present invention under IDV4 fault conditions. Detailed Implementation

[0030] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0031] Reference Figure 1 As shown, this embodiment provides a method for detecting quality-related faults, including: Step S1: Set both process noise and measurement noise to follow a Laplace distribution. Using a Gaussian scale mixture model, reparameterize the Laplace distribution to include both Gaussian and exponential distributions. Establish the dynamic slow characteristic equation, the observation equations for each process variable and the observation equations for each mass variable. Construct a robust probabilistic slow characteristic analysis model and clarify the set of parameters to be optimized in the robust probabilistic slow characteristic analysis model. To better handle environmental noise and outliers, it is assumed that the uncertainty of each variable follows a Laplace distribution. Accordingly, the... process variables and the quality variables In time Robust probabilistic slow feature analysis models include: The dynamic slow characteristic equation is: (1), The observation equation for each process variable is: (2), The observation equation for each mass variable is: (3), in, express Time-of-flight quality-related slow features For the dimensions of slow features related to quality, It is a diagonal weight matrix. For the first A process load vector For the first A mass load vector, Indicates transpose. express Slow characteristics related to time and quality. for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, for Slow-moving noise characteristics for Time of the first Process noise corresponding to each process variable for Time of the first Measurement noise corresponding to each mass variable For process variable indexing, For indexing quality variables, Indicates the time.

[0032] To more accurately represent actual industrial conditions, process noise and measurement noise are modeled as a composite Laplace distribution, with the following formula: (4), (5), in, For the first Process noise corresponding to each process variable For the first Measurement noise corresponding to each mass variable It is a composite Laplace distribution. The first Laplace scaling parameter, This is the second Laplace scale parameter.

[0033] Based on the Gaussian-scale mixture model, the Laplace distribution is reparameterized into a mixture of Gaussian and exponential distributions. Process noise and measurement noise can then be reformulated as a Gaussian-scale mixture, subject to the following distribution constraints: (6), (7), in, For the first Process noise corresponding to each process variable For the first Measurement noise corresponding to each mass variable For process variable indexing, For indexing quality variables, It follows a Gaussian distribution. for Time of the first Gaussian-scaled mixture variables corresponding to each process variable It follows an exponential distribution. The first Laplace scaling parameter, for Time of the first Gaussian-scaled mixture variables corresponding to each quality variable. This is the second Laplacian scaling parameter. Indicates the time.

[0034] Substituting equation (6) into equation (4) and equation (7) into equation (5) respectively, we further obtain: (8), (9),

[0035] in, Let be the probability density function. and Both represent the variance parameters of the conditional Gaussian distribution.

[0036] The parameters to be optimized in the robust probabilistic slow feature analysis model are: The parameter matrix to be solved for the robust probabilistic slow feature analysis model is... For historical process variable data after normalization at different times, The data consists of normalized historical quality variables at different points in time. The slow characteristics related to quality at different times. The process load matrix is ​​composed of all process load vectors. The mass load matrix is ​​composed of all mass load vectors. It is a diagonal weight matrix. The first Laplace scaling parameter, This is the second Laplace scale parameter.

[0037] Step S2: Based on the normalized historical process variable data and quality variable data at different times, determine the parameters of the robust probabilistic slow feature analysis model; In this embodiment, specifically, the method for obtaining normalized historical process variable data and quality variable data at different times includes: Historical operational data from different moments during industrial production are collected. Based on the actual function and representational attributes of the variables, the collected historical data are divided into process variable data that reflect the production operation status. Quality variable data used to characterize the quality of product manufacturing ; This represents the total number of data collection moments, and also the total number of historical samples. The total number of process variables. This represents the total number of quality variables. Subsequently, standardized and normalized preprocessing operations were carried out on the two types of time series data after division. By standardizing the data units and unifying the numerical value range, the modeling interference caused by the differences in units and numerical magnitudes between different variables was eliminated. Simultaneously, basic data denoising preprocessing was completed, effectively removing random noise and invalid redundant information mixed in the data, ensuring that the distribution of all input data is consistent and the data feature expression is consistent. Finally, multi-time historical process variable data and historical quality variable data with time series alignment, uniform specifications and normalization correction were obtained, providing a regular and reliable original input dataset for subsequent offline model training and parameter solving.

[0038] In this embodiment, optionally, the two types of time series data that have been divided are subjected to standardized normalization preprocessing, which means that the two types of time series data are normalized by means so that all variables have zero mean.

[0039] In this embodiment, preferably, the method for determining the parameters of the robust probabilistic slow feature analysis model based on normalized historical process variable data and quality variable data at different times includes: Based on the normalized historical process variable data and quality variable data at different times, the parameters of the robust probability slow feature analysis model are solved with the goal of maximizing the joint probability likelihood function of the robust probability slow feature analysis model.

[0040] Given the process variable and the quality variable, the joint probability likelihood function can be expressed as: (10) in, Let be the joint probability likelihood function. To represent the mass-related slow feature state at the initial moment, Let be the probability density function. The parameter matrix to be solved for the robust probabilistic slow feature analysis model is... For historical process variable data after normalization at different times, The data consists of normalized historical quality variables at different points in time. The slow characteristics related to quality at different times. The process load matrix is ​​composed of all process load vectors. The mass load matrix is ​​composed of all mass load vectors. It is a diagonal weight matrix. The first Laplace scaling parameter, This is the second Laplacian scaling parameter. The total number of moments. The total number of process variables. For the total number of quality variables, for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, express Time-quality related slow features for Time of the first Gaussian-scaled mixture variables corresponding to each process variable for Time of the first Gaussian-scaled mixture variables corresponding to each quality variable. , , It follows an exponential distribution. Indicates time, For process variable indexing, Index for quality variables.

[0041] For simplicity, the optimization objective is optimized, and the methods for determining the final optimization objective include: The joint probability likelihood function of the robust probability slow feature analysis model is transformed into logarithmic form to obtain the logarithmic joint probability likelihood function, with maximizing the logarithmic joint probability likelihood function as the final optimization objective. The log-joint probability likelihood function is: (11), in, Let be the log-joint probability likelihood function.

[0042] Assume that the slow features related to quality follow a Gaussian distribution. The distribution of the remaining terms is given below: (12), (13), (14), (15).

[0043] Methods for solving the parameters of robust probabilistic slow feature analysis models include: S21: Initialize the parameter matrix to be solved for the robust probabilistic slow feature analysis model Initialize the Gaussian scaled mixture variables corresponding to each process variable and quality variable at each time step, and set the current iteration number. and convergence threshold; E-Step: S22: Based on the first The parameter matrix to be solved for the nth iteration, the process noise, measurement noise and corresponding Gaussian-scaled mixture variables at each time step, are obtained by Kalman filtering. The posterior distribution of the quality-related slow features at each time step of the next iteration is calculated, and the expected value of the log joint probability likelihood function is calculated. Using complete mass variable information, perform Kalman forward filtering sequentially: (16), (17), (18), (19), (20), in, Calculated and This will be applied to the subsequent Kalman smoothing process, providing the necessary pre-computed data for its state correction step. To support the correct execution of the forward recursion, their initialization conditions are explicitly stated as follows, encompassing the initial state values ​​and relevant covariance terms: (twenty one), (twenty two), (twenty three), Finally, the number was obtained The posterior distribution of quality-related slow features at each time step of the next iteration .

[0044] in, For the first iteration The prior mean of time-series quality-related slow features. For the first The diagonal weight matrix of the next iteration For the first iteration The posterior mean of time-series quality-related slow features. For the first iteration The posterior mean of time-series quality-related slow features. For the first iteration The prior covariance matrix of time-series quality-related slow features. For the first iteration The posterior covariance matrix of time-series quality-related slow features. For the first The covariance matrix of the noise in the next iteration process For the first iteration The Kalman gain matrix at time 10:00. For the first The process load matrix is ​​composed of all process load vectors from each iteration. For the first The mass load matrix is ​​composed of all mass load vectors from each iteration. It is the identity matrix. For the first iteration The Kalman smoothing gain matrix at time t. For the first iteration The smoothed posterior mean of time-series quality-related slow features. For the first iteration The smoothed posterior mean of time-series quality-related slow features. For the first iteration The smoothed posterior covariance matrix of time-series quality-related slow features.

[0045] S23: Obtain the first... through variational Bayesian inference. The expected values ​​of the Gaussian-scaled mixture variables corresponding to each process variable and the quality variable at each time step of the iteration include: For the iteration At this moment Gaussian-scale mixture variables corresponding to each process variable :definition The formulas for hyperparameters and expectations are: (twenty four), (25), (26), For the iteration At this moment Gaussian-scale mixture variables corresponding to each quality variable :definition The formulas for hyperparameters and expectations are: (27), (28), (29), in, For the first iteration At this moment The expected value of the Gaussian-scaled mixture variables corresponding to each process variable. For the first iteration At this moment The expected value of the Gaussian-scaled mixture variable corresponding to each quality variable. Expressing expectations, For the first The first Laplacian scaling parameter of the next iteration. This is the second Laplacian scaling parameter. for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, Indicates the first iteration In this embodiment, the time-quality related slow feature will be the first... iteration Smoothed posterior mean of time-quality-related slow features As the first iteration Time-quality related slow features For the first A process load vector For the first A mass load vector, Indicates transpose. Indicates time, For process variable indexing, For indexing quality variables, For the first iteration Time of the first The inverse of the implicit scaling variable corresponding to each process variable For the first iteration Time of the first The inverse of the implicit scale variable corresponding to each quality variable It exhibits an inverse gamma distribution.

[0046] S25: Combine the normalized historical process variable data at each time point with the quality variable data, and the... Substituting the expected value of the logarithmic joint probability likelihood function, the expected value of the Gaussian-scaled mixture variable corresponding to each process variable and quality variable at each time step, and the posterior distribution of the quality-related slow features at each time step into the joint probability likelihood function, we can solve for the parameter matrix to be solved, thus obtaining the... The parameter matrix to be solved in the next iteration; Solving for the parameter matrix to be solved yields the first... The parameter matrix to be solved in the nth iteration, the nth The process of solving the parameter matrix in the next iteration includes: Q function definition : Based on the normalized historical process variable data at each time point, the first The posterior distribution of the quality-related slow features at each time step in the iteration, and the expected value of the Gaussian-scaled mixture variable corresponding to each process variable at each time step, are used to calculate the... The process load matrix within the parameter matrix to be solved in the next iteration is given by the following formula: (30), in, For the first iteration Time of the first A process load vector For the first iteration At this moment Gaussian-scaled mixture variables corresponding to each process variable.

[0047] Based on the normalized historical quality variable data at each time point, the first The posterior distribution of the quality-related slow features at each time step in the iteration, and the expected value of the Gaussian-scaled mixture variable corresponding to each quality variable at each time step, are used to calculate the... The mass load matrix within the parameter matrix to be solved in the next iteration is given by the following formula: (31), in, For the first iteration Time of the first A process load vector For the first iteration At this moment Gaussian-scaled mixture variables corresponding to each quality variable.

[0048] Given a diagonal weight matrix where all diagonal elements are greater than or equal to 0 and less than 1, solve for the... The diagonal weight matrix within the parameter matrix to be solved in the next iteration includes: By solving the constructed nonlinear equations, the first... The values ​​of each diagonal element in the diagonal weight matrix within the parameter matrix to be solved in the next iteration; the nonlinear equation is: (32), in, For the first In the diagonal weight matrix of the nth iteration The values ​​of the diagonal elements, Indicates to Find the partial derivative. , For the dimensions of slow features related to quality, The total number of moments. The total number of process variables. For the total number of quality variables, Indicates the first iteration Time-quality related slow features Indicates the first iteration Time-quality related slow features Indicates the time.

[0049] By examining the first The expected values ​​of the Gaussian-scaled mixture variables corresponding to the full-time, full-dimensional process variables in the nth iteration are globally averaged to obtain the nth... The first Laplace scale parameter within the parameter matrix to be solved in the next iteration is given by the following formula: (33), in, For the first The first Laplace scale parameter within the parameter matrix to be solved in the next iteration; By examining the first The expected values ​​of the Gaussian-scaled mixture variables corresponding to the full-time series and full-dimensional quality variables in the nth iteration are globally averaged to obtain the nth... The second Laplace scale parameter within the parameter matrix to be solved in the next iteration is given by the following formula: (34), in, For the first The second Laplace scale parameter within the parameter matrix to be solved in the next iteration.

[0050] S26: Determine the... The parameter matrix to be solved in the nth iteration and the nth iteration The matrix norm of the difference between the parameter matrices to be solved in the next iteration Is it greater than or equal to the convergence threshold? If it is greater than or equal to, then let Return to step S22; otherwise, proceed to step S23. The parameter matrix to be solved in the next iteration As parameters of the robust probabilistic slow feature analysis model.

[0051] This invention employs a hybrid inference strategy that combines the expectation-maximization (EM) algorithm with variational Bayesian inference and Kalman filtering and smoothing. Parameter optimization is achieved iteratively: In the E-step, the posterior distribution of quality-related slow features is estimated using Kalman filtering and smoothing, and the expected values ​​of the Gaussian-scaled mixture variables corresponding to each process variable and quality variable at each time step are updated using variational Bayesian inference; In the M-step, the loading matrix, weight matrix, Laplace scale parameters, etc., are updated by maximizing the logarithmic joint probability likelihood function until the parameters converge. Step S3: Using a robust probabilistic slow feature analysis model, the normalized actual process variable data and quality variable data are compared to obtain statistics for the quality-related slow feature subspace, the quality-related residual subspace, and the quality-independent residual subspace. In this embodiment, quality-oriented monitoring indicators are designed based on three subspaces (quality-related slow feature subspace, quality-related residual subspace, and quality-independent residual subspace) decomposed by the robust probabilistic slow feature analysis model, and corresponding statistics are constructed for each. The steps include: The statistics of the quality-related slow feature subspace are used to monitor slow feature bias. The formula is: (35), in, The quality-related slow features are obtained by using a robust probabilistic slow feature analysis model to compare the normalized actual process variable data and the quality variable data. Let be the covariance matrix of the quality-related slow features estimated from the training data. This refers to the time when the actual variable data was collected. The statistic of the quality-independent residual subspace is used to monitor the reconstruction error of process variables. The formula is: (36), in, For the reconstruction error of process variables, For the reason The process variable vector consisting of all normalized actual process variable data at any given time. The reconstructed process variable values ​​are obtained by reconstructing the process variable vector using the trained robust probabilistic slow feature analysis model. Let be the error covariance matrix of the training data.

[0052] The statistics of the quality-related residual subspace are used to monitor the reconstruction error of quality variables. The formula is: (37), in, For the reconstruction error of process variables, The reconstructed values ​​of the quality variables are obtained by reconstructing the quality variable vector using the trained robust probabilistic slow feature analysis model. Let be the error covariance matrix of the training data.

[0053] Step S4: Based on the statistics of the quality-related slow feature subspace, the quality-related residual subspace, and the quality-independent residual subspace, the distinction between quality-related faults, quality-independent faults, and no faults is realized.

[0054] Kernel density estimation (KDE) is used to model the statistical distribution of monitoring indicators using only normal training data, thus avoiding restrictive distribution assumptions.

[0055] Fault diagnosis should be performed according to the following logic: When monitoring slow characteristic deviation Or monitor the reconstruction error of quality variables When the KDE threshold is exceeded, a quality-related fault is determined to be detected; When monitoring quality variable reconstruction error When the KDE threshold is exceeded, a quality-independent fault is detected. If monitoring slow characteristic deviation Monitoring quality variable reconstruction error Monitoring process variable reconstruction error None of them exceeded their KDE thresholds, so they were determined to be fault-free.

[0056] This invention constructs a quality-oriented monitoring index system and a kernel density estimation threshold determination method based on three subspaces, which enables accurate differentiation between quality-related faults and quality-independent faults. It is also adapted to actual industrial scenarios where quality variables cannot be measured online, effectively reducing false alarms and missed detections, reducing unnecessary production downtime losses, and ensuring product quality stability and production process safety.

[0057] If the method of this invention is applied to the monitoring of continuous fermentation production in industrial chemical industry, the process variables include on-site operating parameters such as temperature inside the fermenter, pressure inside the fermenter, pH value of the culture medium, stirring speed, aeration flow rate, material feeding rate, and dissolved oxygen content in the culture medium; the quality variables include indicators characterizing the quality of the fermented product such as the number of viable bacteria in fermentation, concentration of the target product, purity of fermentation broth, product conversion rate, and residual sugar content in fermentation. The model building, parameter determination, and online fault identification can be completed based on these two types of variables.

[0058] Based on Example 1, this Example 2 uses the Tennessee-Eastman (TE) process as an example. The TE process is a widely used chemical process simulation model for testing and validating control and monitoring strategies in industrial applications. This model incorporates complex interactions between multiple units and various chemical substances, simulating real-world industrial scenarios. Twenty-two measured variables and eleven manipulated variables were selected to define the process variable space X, while one measured variable was chosen as the quality variable space Y. In the TE process, 500 normal samples were collected as a training dataset, and a test dataset containing 160 normal samples and 800 failure samples was prepared.

[0059] To verify the effectiveness of the proposed quality-related fault detection method (RPQSFA) in industrial process fault detection, the model was tested using methods such as PPCR, PLVR, OPLS, and MPPLS.

[0060] As shown in Table 1, Table 1 presents the root mean square error (RMSE) and coefficient of determination (CQSFA) of PPCR, PLVR, OPLS, MPPLS, and RPQSFA under different TE faults. ).

[0061] Table 1

[0062] like Figure 2 As shown, Figure 2 This is a graph showing the prediction performance of this invention and other algorithms under IDV13 fault conditions. Figure 2 From top to bottom, the models correspond to PPCR, MPPLS, PLVR, and RPQSFA, respectively.

[0063] From Table 1 and Figure 2 It can be seen that, with an outlier rate of 5%, the RPQSFA method has the lowest RMSE. The highest value indicates that the RPQSFA method has good prediction accuracy and robustness. For example, as... Figure 2 As shown, the RPQSFA method maintains stable and accurate predictions even in the presence of outliers, while other methods exhibit significant bias.

[0064] As shown in Table 2, Table 2 presents the monitoring results of quality-related failures using the experimental factor level table methods of PPCR, PLVR, OPLS, MPPLS, and RPQSFA.

[0065] Table 2

[0066] As shown in Table 3, Table 3 presents the monitoring results of quality-independent faults using the experimental factor level table methods of PPCR, PLVR, OPLS, MPPLS, and RPQSFA.

[0067] Table 3

[0068] like Figure 3 , Figure 4 , Figure 5 , Figure 6 As shown, Figure 3 This is a graph showing the monitoring performance of the RRCR algorithm under IDV13 fault conditions. Figure 4 This is a graph showing the monitoring performance of the RLVR algorithm under IDV13 fault conditions. Figure 5 This is a graph showing the monitoring performance of the MPPLS algorithm under IDV13 fault conditions. Figure 6 This is a monitoring performance diagram of the present invention under IDV13 fault conditions.

[0069] like Figure 7 , Figure 8 , Figure 9 , Figure 10 As shown, Figure 7 This is a graph showing the monitoring performance of the RRCR algorithm under IDV4 fault conditions. Figure 8 This is a graph showing the monitoring performance of the RLVR algorithm under IDV4 fault conditions. Figure 9 This is a graph showing the monitoring performance of the MPPLS algorithm under IDV4 fault conditions. Figure 10 This is a graph showing the monitoring performance of the present invention under IDV4 fault conditions.

[0070] Among them, IDV1, IDV2, IDV4, IDV6, IDV8, IDV11, IDV13, and IDV14 are all fault numbers of the TE process.

[0071] From Table 2, Table 3 and Figure 2 , Figure 3 It can be seen that, in terms of fault detection performance for quality-related faults, the RPQSFA, PPCR, PLVR, and MPPLS methods can all correctly identify the fault type, i.e., diagnose it as a quality-related fault, based on their respective detection logic. For most faults, these four methods... The FDRs of the statistics all exceeded 90%, and some even reached 100%. This is the Hotling statistic. Furthermore... Figure 3 , Figure 4 , Figure 5 , Figure 6 It is clearly shown that, for fault IDV13, all four methods can accurately detect the fault from its inception. Notably, Table 2 illustrates the performance of the proposed RPQSFA method. The statistical FDRs reached 100%, which is superior to the PPCR, PLVR, and MPPLS methods. The statistics clearly demonstrate the superior performance of the proposed method.

[0072] For quality-independent faults such as IDV4, IDV11, and IDV14, the proposed RPQSFA method can be observed to perform well in the quality-related subspace. and The statistic exhibits lower FDRs in the quality-correlated subspace, but lower FDRs in the quality-independent subspace. The statistic shows high FDRs. Therefore, based on its fault detection logic, the RPQSFA method accurately classifies these faults as quality-independent faults. In contrast, the PPCR method produces relatively high FDRs in the quality-related subspace using the Q-statistic. Specifically, the FDRs for IDV11 and IDV14 exceed 40%, significantly higher than the RPQSFA method, contradicting the actual fault characteristics. Furthermore, for IDV4 and IDV11 faults, the Q-statistic FDRs of PPCR in the quality-independent subspace are only around 40%, significantly lower than the FDRs of RPQSFA in the corresponding quality-independent subspace. Therefore, the RPQSFA method is superior to the PPCR method.

[0073] Further testing of the PLVR and MPPLS methods revealed that although the FDRs of most faults in the quality-related subspace were lower than those in the PPCR, they still generally exceeded 16%. Notably, for fault IDV14, the PLVR quality-related subspace showed... The FDR of the statistics even exceeds 70%, which can easily lead to misclassification of fault types.

[0074] In summary, the proposed RPQSFA method benefits from the Laplace distribution noise assumption in the model and incorporates dynamic characteristics when extracting common latent variables from quality and process variables. Therefore, it exhibits superior prediction and fault detection performance.

[0075] This third embodiment provides a quality-related fault detection system, including: The model building module is used to set both process noise and measurement noise to follow a Laplace distribution. Using a Gaussian scale mixture model, the Laplace distribution is reparameterized into a form that includes both Gaussian and exponential distributions. The dynamic slow feature equation, the observation equations for each process variable and the observation equations for each mass variable are established to build a robust probabilistic slow feature analysis model. The parameter determination module is used to determine the parameters of the robust probabilistic slow feature analysis model based on the normalized historical process variable data and quality variable data at different times. The statistics module is used to obtain statistics for the quality-related slow feature subspace, quality-related residual subspace, and quality-independent residual subspace by using a robust probabilistic slow feature analysis model to analyze the normalized actual process variable data and quality variable data. The identification module is used to distinguish between quality-related faults, quality-independent faults, and no faults based on statistics of the quality-related slow feature subspace, the quality-related residual subspace, and the quality-independent residual subspace.

[0076] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0077] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0078] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0079] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0080] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A method for detecting quality-related faults, characterized in that, include: Assuming that both process noise and measurement noise follow a Laplace distribution, a Gaussian scale mixture model is used to reparameterize the Laplace distribution into a form that includes both Gaussian and exponential distributions. Dynamic slow characteristic equations, observation equations for each process variable, and observation equations for each mass variable are established, and a robust probabilistic slow characteristic analysis model is constructed. Based on the normalized historical process variable data and quality variable data at different times, the parameters of the robust probabilistic slow feature analysis model are determined. The normalized actual process variable data and quality variable data are used to obtain statistics for the quality-related slow feature subspace, quality-related residual subspace, and quality-independent residual subspace through a robust probabilistic slow feature analysis model. Based on the statistics of the quality-related slow feature subspace, the quality-related residual subspace, and the quality-independent residual subspace, the system can distinguish between quality-related faults, quality-independent faults, and no faults.

2. The method for detecting quality-related faults according to claim 1, characterized in that, After reparameterizing the Laplace distribution to include both Gaussian and exponential distributions, the process noise and measurement noise obey the following distribution constraints: ， ， in, For the first Process noise corresponding to each process variable For the first Measurement noise corresponding to each mass variable For process variable indexing, For indexing quality variables, It follows a Gaussian distribution. for Time of the first Gaussian-scaled mixture variables corresponding to each process variable It follows an exponential distribution. The first Laplace scaling parameter, for Time of the first Gaussian-scaled mixture variables corresponding to each quality variable. This is the second Laplacian scaling parameter. Indicates the time.

3. The quality-related fault detection method according to claim 1, characterized in that, The dynamic slow characteristic equation is: ， The observation equation for each process variable is: ， The observation equation for each mass variable is: ， in, express Time-quality related slow features For the dimensions of slow features related to quality, It is a diagonal weight matrix. For the first A process load vector For the first A mass load vector, Indicates transpose. express Slow characteristics related to time and quality. for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, for Slow-moving noise characteristics for Time of the first Process noise corresponding to each process variable for Time of the first Measurement noise corresponding to each mass variable For process variable indexing, For indexing quality variables, Indicates the time.

4. The quality-related fault detection method according to claim 1, characterized in that, Methods for determining the parameters of a robust probabilistic slow feature analysis model based on normalized historical process variable data and quality variable data at different times include: Based on the normalized historical process variable data and quality variable data at different times, the parameters of the robust probability slow feature analysis model are solved with the goal of maximizing the joint probability likelihood function of the robust probability slow feature analysis model.

5. The quality-related fault detection method according to claim 4, characterized in that, Methods for optimizing the objective and determining the final optimization objective include: The joint probability likelihood function of the robust probability slow feature analysis model is transformed into logarithmic form to obtain the logarithmic joint probability likelihood function, with maximizing the logarithmic joint probability likelihood function as the final optimization objective. The log-joint probability likelihood function is: ， in, Let be the log-joint probability likelihood function. Let be the probability density function. The parameter matrix to be solved for the robust probabilistic slow feature analysis model is... For historical process variable data after normalization at different times, The data consists of historical quality variables after normalization at different times. The slow characteristics related to quality at different times. The process load matrix is ​​composed of all process load vectors. The mass load matrix is ​​composed of all mass load vectors. It is a diagonal weight matrix. The first Laplace scaling parameter, This is the second Laplacian scaling parameter. The total number of moments. The total number of process variables. For the total number of quality variables, for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, express Time-quality related slow features for Time of the first Gaussian-scaled mixture variables corresponding to each process variable for Time of the first Gaussian-scaled mixture variables corresponding to each quality variable. , , It follows an exponential distribution. Indicates time, For process variable indexing, Index for quality variables.

6. The quality-related fault detection method according to claim 5, characterized in that, Methods for solving the parameters of robust probabilistic slow feature analysis models include: S21: Initialize the parameter matrix to be solved for the robust probabilistic slow feature analysis model, initialize the Gaussian scale mixture variables corresponding to each process variable and quality variable at each time step, and set the current iteration number. and convergence threshold; S22: Based on the first The parameter matrix to be solved for the nth iteration, the process noise, measurement noise and corresponding Gaussian-scaled mixture variables at each time step, are obtained by Kalman filtering. The posterior distribution of the quality-related slow features at each time step of the next iteration is calculated, and the expected value of the log joint probability likelihood function is calculated. S23: Obtain the first... through variational Bayesian inference. The expected value of the Gaussian-scaled mixture variable corresponding to each process variable and quality variable at each time point of each iteration; S24: Combine the normalized historical process variable data and the quality variable data at each time point, and... Substituting the expected value of the logarithmic joint probability likelihood function, the expected value of the Gaussian-scaled mixture variable corresponding to each process variable and quality variable at each time step, and the posterior distribution of the quality-related slow features at each time step into the joint probability likelihood function, we solve for the parameter matrix to be solved, and obtain the... The parameter matrix to be solved in the next iteration; S25: Determine the... The parameter matrix to be solved in the nth iteration and the nth iteration If the matrix norm of the difference between the parameter matrices to be solved in the next iteration is greater than or equal to the convergence threshold, then let... Return to step S22; otherwise, proceed to step S23. The parameter matrix to be solved in the next iteration is used as the parameter of the robust probabilistic slow feature analysis model.

7. The quality-related fault detection method according to claim 6, characterized in that, Obtain the first... The formula for calculating the expected value of the Gaussian-scaled mixture variable corresponding to each process variable and the mass variable at each time step of the iteration is as follows: ， ， in, For the first iteration At this moment The expected value of the Gaussian-scaled mixture variables corresponding to each process variable. For the first Second iteration At this moment The expected value of the Gaussian-scaled mixture variable corresponding to each quality variable. Expressing expectations, , , For the first The first Laplacian scaling parameter of the next iteration. This is the second Laplacian scaling parameter. for Time of the first The values ​​of process variables, for Time of the first The values ​​of the quality variables, Indicates the first Second iteration The posterior distribution of time-quality-related slow features. For the first A process load vector For the first A mass load vector, Indicates transpose. Indicates time, For process variable indexing, Index for quality variables.

8. The quality-related fault detection method according to claim 6, characterized in that, Solving for the parameter matrix to be solved yields the first... The parameter matrix to be solved in the nth iteration, the nth The process of solving the parameter matrix in the next iteration includes: Based on the normalized historical process variable data at each time point, the first The posterior distribution of the quality-related slow features at each time step in the iteration, and the expected value of the Gaussian-scaled mixture variable corresponding to each process variable at each time step, are used to calculate the... The process load matrix within the parameter matrix to be solved in the next iteration; Based on the normalized historical quality variable data at each time point, the first The posterior distribution of the quality-related slow features at each time step in the iteration, and the expected value of the Gaussian-scaled mixture variable corresponding to each quality variable at each time step, are used to calculate the... The mass load matrix within the parameter matrix to be solved in the next iteration; Given a diagonal weight matrix where all diagonal elements are greater than or equal to 0 and less than 1, solve for the... The diagonal weight matrix within the parameter matrix to be solved in the next iteration; By examining the first The expected values ​​of the Gaussian-scaled mixture variables corresponding to the full-time, full-dimensional process variables in the nth iteration are globally averaged to obtain the nth... The first Laplace scale parameter within the parameter matrix to be solved in the next iteration; By examining the first The expected values ​​of the Gaussian-scaled mixture variables corresponding to the full-time series and full-dimensional quality variables in the nth iteration are globally averaged to obtain the nth... The second Laplace scale parameter within the parameter matrix to be solved in the next iteration.

9. A method for detecting quality-related faults according to claim 8, characterized in that, Solve the first The methods for solving the diagonal weight matrix within the parameter matrix in the next iteration include: By solving the constructed nonlinear equations, the first... The values ​​of each diagonal element in the diagonal weight matrix within the parameter matrix to be solved in the next iteration; the nonlinear equation is: ， in, For the first In the diagonal weight matrix of the nth iteration The values ​​of the diagonal elements, Indicates to Find the partial derivative. , For the dimensions of slow features related to quality, The total number of moments. The total number of process variables. For the total number of quality variables, Indicates the first Second iteration Time-quality related slow features Indicates the first Second iteration Time-quality related slow features Indicates the time.

10. A quality-related fault detection system, characterized in that, include: The model building module is used to set both process noise and measurement noise to follow a Laplace distribution. Using a Gaussian scale mixture model, the Laplace distribution is reparameterized into a form that includes both Gaussian and exponential distributions. The dynamic slow feature equation, the observation equations for each process variable and the observation equations for each mass variable are established to build a robust probabilistic slow feature analysis model. The parameter determination module is used to determine the parameters of the robust probabilistic slow feature analysis model based on the normalized historical process variable data and quality variable data at different times. The statistics module is used to obtain statistics for the quality-related slow feature subspace, quality-related residual subspace, and quality-independent residual subspace by using a robust probabilistic slow feature analysis model to analyze the normalized actual process variable data and quality variable data. The identification module is used to distinguish between quality-related faults, quality-independent faults, and no faults based on statistics of the quality-related slow feature subspace, the quality-related residual subspace, and the quality-independent residual subspace.

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