Robust fault detection method and system for outliers non-stationary industrial processes

Abstract

The application provides a robust fault detection method and system for outliers and non-stationary industrial processes, which comprises the following steps: obtaining multi-sensor and multi-variable historical operation data under normal working conditions, standardizing the data, dividing the data into continuous and non-overlapping time periods according to time, and introducing time period indicating variables; constructing a robust probability stationary subspace analysis model, decomposing observation samples into process components, process noise and outlier vectors, and distinguishing stationary sources from non-stationary sources; modeling the heavy tail of potential sources by using student t distribution and Gaussian-Gamma mixed form, introducing scale variables, mixed weights and prior distribution; training the model by using a variational Bayesian expectation maximization algorithm, updating the posterior of hidden variables and model parameters; determining control limits based on stationary source statistics and process uncertainty sliding window Wasserstein distance statistics, and performing fault judgment and alarm on online samples, which is suitable for complex scenes where outliers and non-stationarity coexist, and improves detection stability and accuracy.

Description

Technical Field

[0001] This invention relates to the field of industrial process automatic control technology, and in particular to a robust fault detection method and system for outlier non-stationary industrial processes. Background Technology

[0002] Real-world industrial processes are typically not strictly stationary. They are usually affected by factors such as fluctuations in raw material properties, changes in load, catalyst deactivation, equipment aging, environmental disturbances, and adjustments to operating conditions. The mean, variance, and correlation structure of process variables change over time. For these non-stationary processes, traditional static monitoring methods such as principal component analysis and partial least squares analysis are prone to misinterpreting normal operating condition drift as faults, or masking early fault characteristics with non-stationary trends, leading to increased false alarm rates or delayed fault detection.

[0003] For monitoring non-stationary processes, existing methods mainly include model adaptive updating, cointegration analysis, and stationary subspace analysis. Model adaptive updating recursively updates the model to adapt to process changes; however, as a fault gradually develops, the model may simultaneously adapt to the fault, causing a delay in fault detection. Cointegration analysis extracts long-term equilibrium relationships between non-stationary variables for monitoring, but it relies heavily on the cointegration assumption, which is often complex and easily affected by disturbances in actual industrial processes, making it difficult to satisfy this assumption in the long run. Stationary subspace analysis decomposes observed data into stationary and non-stationary components, which can mitigate the impact of non-stationary trends on monitoring results to some extent; further, probabilistic stationary subspace analysis can probabilistically model process uncertainties, thereby distinguishing between normal process changes and random noise.

[0004] However, the aforementioned non-stationary process monitoring methods typically assume that the modeling data is relatively "clean" and do not fully consider outliers commonly found in actual industrial data. In industrial settings, transient interference, short-term measurement spikes, communication anomalies, occasional sensor malfunctions, or localized operational disturbances can all generate isolated anomalous observations. These outliers usually occur infrequently but can be quite significant, disproportionately impacting the parameter estimation of data-driven models and easily causing the model to deviate from its portrayal of normal process behavior.

[0005] In non-stationary industrial processes, the impact of outliers is more subtle. Non-stationary trends can mask the abrupt changes of outliers, making them difficult to identify directly; conversely, outliers can distort the estimation of normal operating conditions by non-stationary models. In the offline modeling stage, outliers mixed into the training data may lead to biases in the estimation of the mixing matrix, source distribution parameters, or noise parameters, causing a mismatch in the monitoring model. In the online monitoring stage, the model may misjudge normal non-stationary fluctuations as faults, or it may drown out real faults under non-stationary trends and abnormal disturbances, resulting in a decrease in monitoring reliability.

[0006] Therefore, there is a need to provide a robust fault detection method and system for non-stationary industrial processes with outliers. This method should not only achieve the decomposition of stationary / non-stationary sources, but also robustly suppress the influence of outliers on model parameters, and jointly monitor anomalies in stationary components and changes in the distribution of process uncertainty, thereby improving the stability and accuracy of fault detection in complex industrial processes. Summary of the Invention

[0007] To address the aforementioned shortcomings of the existing technology, the purpose of this invention is to provide a robust fault detection method and system for outlier non-stationary industrial processes, aiming to solve technical problems such as easy bias in the modeling of outlier non-stationary industrial processes and high false alarm and false negative rates.

[0008] To achieve the above objectives, this invention provides a robust fault detection method for non-stationary industrial processes with outliers, comprising the following steps: The system acquires multivariate historical operating data collected by multiple sensors in an industrial process control system under normal operating conditions. After standardization, the data is divided into multiple continuous non-overlapping time periods, and time period indicator variables are introduced. A robust probabilistic stationary subspace analysis model is constructed by combining time-period indicator variables. The observed sample is represented as the sum of process components, process noise, and outlier vectors after the mixture matrix is ​​applied to the potential source. The potential source includes stationary and non-stationary sources. The mean and covariance of the stationary source are shared in each time period, while the mean and covariance of the non-stationary source change with time period. The student t-distribution is used to model the potential sources with heavy tails. A robust probability model is formed by introducing a scale variable, a time period indicator variable, a mixture weight, a Dirichlet prior of the mixture weight, and a gamma prior of the degrees of freedom through a mixture of Gaussian and gamma distributions. The robust probability model is trained using the variational Bayesian expectation-maximization algorithm, and the posterior distribution of each latent variable, as well as the mixture matrix, process noise variance, mean and covariance of stationary and non-stationary sources, are updated to obtain the trained model. The stationary source statistic of the training samples and the Wasserstein distance statistic, determined by the empirical mean and empirical covariance of the sliding window of the process uncertainty estimate, are calculated based on the training model, and the corresponding control limits are determined. Input the online running samples into the training model, calculate the online stationary source statistic and the online Wasserstein distance statistic. If either statistic exceeds the corresponding control limit, a fault is determined and an alarm message is output.

[0009] As a further improvement to the above technical solution, the multivariate historical operating data includes process variables collected by at least two types of sensors, including temperature sensors, pressure sensors, flow sensors, material quantity detection devices, gas parameter detection devices, and equipment operating status detection devices. When standardizing the multivariate historical operating data, the mean and standard deviation of each process variable under normal operating conditions are used to standardize the historical samples and the online operating samples using the same parameters. The time period indicator variable satisfies: ; in, Indicates the first Does the observed sample belong to the first...? Each period, Indicates belonging to the first Each period, Indicates that it does not belong to the first Each period, This indicates the number of consecutive, non-overlapping time periods.

[0010] As a further improvement to the above technical solution, the robust probabilistic stationary subspace analysis model satisfies: ; in, In the formula, For the first Standardized observation samples at each sampling time, It is an invertible mixture matrix. Let be a submatrix of a Spanning stationary subspace. Let be a submatrix of a nonstationary subspace. For potential source vectors, For a stable source, It is a non-stationary source. For process noise, The vector of outliers, This represents the transpose of a vector.

[0011] As a further improvement to the above technical solution, the process noise satisfies: ; The dimension of the stationary source is The dimension of the non-stationary source is And satisfy: ; In the formula, Indicates process noise. Indicates a Gaussian distribution. Represents the process noise variance. express 3D identity matrix The dimension of the process variable representing the observed sample. Denotes the dimension of the stationary source. express 3D real space, express dimensional real space; the dimension of the stationary source This was determined using the Johansen cointegration test.

[0012] As a further improvement to the above technical solution, in the first Within a given time period, the mean and covariance of the stationary source are shared across all time periods, and the specific variations of the mean and covariance of the non-stationary source over time satisfy the following: ; In the formula, Indicates the first The mean of potential sources over a given period, Indicates the first Potential source covariance over time period This represents the mean of a stationary source. Indicates the covariance of stationary sources. Indicates the first The mean of non-stationary sources over a given period Indicates the first The non-stationary source covariance for each time period, with zero elements in the matrix indicating that there is no correlation between the stationary and non-stationary source groups.

[0013] As a further improvement to the above technical solution, the heavy-tailed modeling of the potential source by the student t-distribution is specifically as follows: ; It can be represented as a mixture of Gaussian and gamma distributions: ; In the formula, Indicates the first The probability density of potential sources within a given time period. Indicates the first The observation sample at the ... Potential sources corresponding to each time period Describe the t-distribution of students. Indicates the first The mean of potential sources over a given period, Indicates the first Potential source covariance over time period Indicates degrees of freedom. Represents a scale variable. Indicates a Gaussian distribution. The scale variable represents the gamma distribution; when the observed sample deviates from the potential source distribution of its time period, the scale variable is used to reduce the impact of the observed sample on the model parameter update.

[0014] As a further improvement to the above technical solution, the Dirichlet prior of the hybrid weights satisfies: ; The gamma prior of the degrees of freedom is satisfied as follows: ; In the formula, Represents the mixed weight vector. Indicates that the observed sample belongs to the first Prior weights for each time period, Indicates the Dirichlet distribution. Denotes the Dirichlet prior parameters. Let be the normalization constant (standardization coefficient) of the Dirichlet distribution, ensuring that the integral over the entire space equals 1; For an n-fold multiplication operator, iterate through all n time periods / mixed components, where n is the total number of time periods; exponent By hyperparameters Decide and control the i-th weight The prior distribution shape; This represents the degrees of freedom of the student's t-distribution. Represents the gamma distribution. and Let represent the shape parameter and rate parameter of the prior gamma distribution of the degrees of freedom, respectively.

[0015] As a further improvement to the above technical solution, the Variational Bayesian Expectation-Maximization (VBEM) algorithm divides the variables to be updated into a latent variable set and a deterministic parameter set; the latent variable set is: ; The deterministic parameter set is as follows: ; In the formula, Represents a set of latent variables. Indicates a time period indicator variable. Indicates a potential source. Represents a scale variable. Indicates degrees of freedom. Indicates mixed weights; Represents a deterministic set of parameters. This represents the mean of a stationary source. This represents the mean of a non-stationary source. Indicates the covariance of stationary sources. This represents the covariance of a non-stationary source. Represents a mixture matrix. The variance of process noise is represented; the variational Bayesian expectation-maximization algorithm alternately executes the variational Bayesian expectation step (VBE step) and the variational Bayesian maximum step (VBM step), wherein the VBE step is used to update the posterior distribution of the latent variable group, and the VBM step is used to update the deterministic parameter group.

[0016] As a further improvement to the above technical solution, in the VBE step, the potential source The posterior distribution is: ; in, ; ; In the formula, Let represent the optimal variational posterior distribution of the potential source. This represents the posterior mean of the potential source. This represents the potential source posterior covariance. Indicates a Gaussian distribution. Represents a mixture matrix. Represents the process noise variance. Indicates a time period indicator variable. Represents a scale variable. Indicates the first Potential source covariance over time period Indicates the first The mean of potential sources over a given period, Indicates the first One observation sample, Indicates the number of time periods. This represents the transpose of a vector. This represents the expectation under the corresponding variational distribution.

[0017] As a further improvement to the above technical solution, in the VBE step, the scale variable... The posterior distribution is: ; in, ; ; In the formula, This represents the optimal variational posterior distribution of the scaling variable. Represents the gamma distribution. The shape parameter representing the posterior distribution of the scaling variable. The rate parameter representing the posterior distribution of the scaling variable. Indicates degrees of freedom. Indicates the dimension of the process variable. Indicates the number of time periods. Indicates a time period indicator variable. Indicates a potential source. Indicates the first The mean of potential sources over a given period, Indicates the first Potential source covariance over time period This represents the transpose of a vector. This represents the expectation under the corresponding variational distribution.

[0018] As a further improvement to the above technical solution, the stationary source statistic is: ; in, ; In the formula, Indicates the first The stationary source statistic for each sample. Indicates the first Potential source estimates for each sample Indicates the first Stationary source estimates for each sample. This represents the stationary source selection matrix. This represents the mean of the latent source estimates during the training phase. This indicates the covariance of the potential source estimates during the training phase. Indicates vector transpose; The uncertainty estimate for the process is: ; Use a length of Uncertainty estimates for the sliding window collection process: ; And calculate the Wasserstein distance statistic according to the following formula: ; In the formula, Indicates the first Estimated process uncertainty for a single sample Indicates the first Observations of a sample Represents a mixture matrix. Indicates the first Potential source estimates for each sample This represents the set of process uncertainty estimates within the sliding window. Indicates the length of the sliding window. Indicates the first Wasserstein distance statistic for each sample This represents the empirical mean of the process uncertainty estimates within the sliding window. The empirical covariance represents the estimate of process uncertainty within the sliding window. Indicates the dimension of the process variable. Indicates the standard deviation of process noise. Represents the trace of a matrix. This represents the L2 norm.

[0019] As a further improvement to the above technical solution, the stationary source statistic and Wasserstein distance statistic calculated using training samples are used to estimate the kernel density at a pre-set confidence level. Determine the control limits for stationary source statistics Control limits for Wasserstein distance statistic ; During online monitoring, the mixture matrix, process noise variance, mean of stationary sources, covariance of stationary sources, mean of non-stationary sources, and covariance of non-stationary sources in the fixed training model are inferred only for the time period indicator variables, potential sources, and scale variables corresponding to the online running samples. Fault determination meets the following criteria: ; In the formula, Indicates the preset credit level. This indicates the control limits for stationary source statistics. This indicates the control limits for the Wasserstein distance statistic. This represents the online stationary source statistics. This represents the online Wasserstein distance statistic.

[0020] Secondly, the present invention also provides a robust fault detection system for outlier nonstationary industrial processes, including industrial process sensors, a data processing server, and an alarm terminal. The industrial process sensor is used to collect multivariable historical operating data and online operating samples of the industrial process control system under normal operating conditions. The data processing server includes a processor and a memory, wherein the memory stores a program executable by the processor, and when the program is executed by the processor, it implements the robust fault detection method described in the first aspect. The alarm terminal is used to receive fault alarm information output by the data processing server and display the normal or fault status of the industrial process.

[0021] Thirdly, the present invention also provides an electronic device, including a processor, a memory, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it is used to implement the steps of the robust fault detection method for outlier non-stationary industrial processes described in the first aspect.

[0022] Fourthly, the present invention also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, is used to implement the steps of the robust fault detection method for outlier non-stationary industrial processes described in the first aspect.

[0023] Because the present invention adopts the above technical solutions, the beneficial effects of the present invention are as follows:

[0024] This invention provides a robust fault detection method for non-stationary industrial processes with outliers. Addressing the issues of biased normal-condition models and false alarms / false negatives in fault detection when non-stationary trends and outlier disturbances coexist in actual industrial processes, this invention achieves robust modeling and online fault detection for non-stationary industrial processes containing outliers through a combination of time-varying non-stationary modeling, Student's t-distribution heavy-tailed modeling, variational Bayesian joint solution, and dual statistical monitoring. Compared with existing technologies, this invention has at least the following advantages:

[0025] First, this invention divides multivariate historical operating data collected under normal operating conditions into multiple continuous, non-overlapping time periods and introduces time period indicator variables to characterize the phased changes in the statistical characteristics of industrial processes. Based on this, the mean and covariance of stationary sources are shared across time periods, while the mean and covariance of non-stationary sources vary with time. Therefore, the model can describe non-stationary trends under normal operating conditions while preserving stable process relationships, avoiding the simplistic interpretation of normal statistical changes caused by load fluctuations, operating condition adjustments, or slow equipment changes as faults, thus helping to reduce false alarms under non-stationary operating conditions.

[0026] Second, this invention constructs a robust probabilistic stationary subspace analysis model, representing the observed samples as the sum of process components, process noise, and outlier vectors after a mixing matrix is ​​applied to the potential sources, and classifying potential sources into stationary and non-stationary sources. Through this modeling approach, normal non-stationary changes and stable process relationships are described separately. Fault detection no longer relies solely on the overall shift of the original observed variables but can further focus on anomalous changes in stable relationships within the stationary subspace. Therefore, when fault changes are partially masked by non-stationary trends, this invention can still detect process state deviations based on stationary source statistics, improving the reliability of fault detection.

[0027] Third, this invention employs the Student's t-distribution for heavy-tailed modeling of potential sources and introduces a scaling variable through a hybrid of Gaussian and gamma distributions. Because the Student's t-distribution is heavy-tailed, a small number of outliers with large amplitudes will not have the same excessive impact on model parameter estimation as under the ordinary Gaussian assumption. Furthermore, the scaling variable's participation in posterior updates reduces the influence weight of samples deviating from their respective time periods in parameter updates. Therefore, even with transient interference, measurement spikes, or occasional sensor anomalies in the training data, the model parameters are less likely to be skewed by outliers, thus improving the robustness of the model under normal operating conditions.

[0028] Fourth, this invention employs a variational Bayesian expectation-maximization algorithm to train a robust probabilistic model. It uses time-period indicator variables, latent sources, scale variables, degrees of freedom, and mixture weights as latent variables for posterior inference, and simultaneously updates the mixture matrix, process noise variance, and the means and covariances of stationary and non-stationary sources. Since there is a coupling relationship between time-period attribution, outlier influence, latent source estimation, and model parameters, this invention uses a unified probabilistic inference framework for alternating updates, which reduces error propagation caused by independent step-by-step estimation and makes the model training process more stable.

[0029] Fifth, this invention simultaneously employs stationary source statistics and Wasserstein distance statistics. The stationary source statistics monitor deviations from stable relationships within the stationary subspace, suitable for reflecting changes in process coupling relationships or operational mechanisms when abnormalities occur. The Wasserstein distance statistics, determined based on the empirical mean and empirical covariance of a sliding window of process uncertainty estimates, measure the shift in the process uncertainty distribution relative to the normal state. These two types of statistics evaluate the operational status from two aspects: changes in stationary components and changes in the process uncertainty distribution. This allows for the coverage of more types of abnormal behavior and reduces the risk of missed detections due to relying on a single statistic.

[0030] Sixth, this invention determines the control limits of stationary source statistics and Wasserstein distance statistics based on training samples, and inputs online operating samples into the training model to calculate the corresponding statistics during online monitoring. When any statistic exceeds the corresponding control limit, a fault is determined and an alarm message is output. This determination method has clear rules and is easy to implement in industrial process control systems; at the same time, the online monitoring stage is based on the normal operating condition model obtained from the training, avoiding the continuous absorption of fault samples into the model update process, thus helping to maintain the sensitivity and consistency of fault alarms.

[0031] In summary, this invention, through a combination of techniques including "time-period indicator variables describing non-stationary phase changes, stationary / non-stationary source decomposition representing process structure, Student's t-tailed distribution suppressing outlier influence, variational Bayesian algorithm jointly training the model, and dual statistics for online monitoring," enables the model to simultaneously adapt to industrial data scenarios where non-stationary trends, process noise, and outlier interference coexist. This approach improves the stability of normal operating condition modeling and provides a more reliable basis for online fault detection in complex industrial processes. Attached Figure Description

[0032] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0033] Figure 1 This is a flowchart illustrating a robust fault detection method for outlier nonstationary industrial processes disclosed in this invention. Figure 2 It is the probabilistic directed graph of the robust probabilistic stationary subspace analysis model disclosed in this invention; Figure 3 This is a schematic diagram of the process flow of a certain industrial fluidized bed roasting process disclosed in this invention. Figure 4 This is a comparison chart of monitoring results of different fault detection methods disclosed in this invention on industrial roasting process data, wherein: Figure 4 (a) is a graph showing the monitoring results of the T² statistic using the RPCA method; Figure 4 (b) is a graph showing the monitoring results of the SPE statistic using the RPCA method; Figure 4 (c) is a graph showing the monitoring results of the T² statistic using the SSA method; Figure 4 (d) is a graph showing the monitoring results of the T² statistic using the EASSA method; Figure 4 (e) represents the PSSA method. Statistical monitoring results chart; Figure 4 f is the PSSA method Statistical monitoring results chart; Figure 4 (g) is the RPSSA method of the present invention. Statistical monitoring results chart; Figure 4 (h) is the WD of the RPSSA method of the present invention. e Statistical monitoring results chart.

[0034] Figure label: 1. Feeding unit; 2. Blowering unit; 3. Roasting furnace; 4. Exhaust gas unit; 5. Roasting unit.

[0035] The realization of the objective, functional characteristics and advantages of the present invention will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0036] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0037] It should be noted that the technical solutions of the various embodiments of the present invention can be combined with each other, but only if they are based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0038] Example 1

[0039] See Figure 1 and Figure 2 This invention provides a robust fault detection method for outlier, non-stationary industrial processes. The method mainly includes an offline modeling stage and an online monitoring stage. The offline modeling stage is used to establish a robust probabilistic model based on historical data from normal operating conditions and determine monitoring control limits. The online monitoring stage is used to determine the status of real-time operating samples and output alarm information.

[0040] First, multivariate historical operating data collected by multiple sensors in the industrial process control system under normal operating conditions is acquired. This multivariate historical operating data includes process variables collected by at least two types of sensors, including temperature sensors, pressure sensors, flow sensors, material quantity detection devices, gas parameter detection devices, and equipment operating status detection devices. To reduce the impact of different dimensions on subsequent model training, each process variable is standardized. Specifically, historical samples can be standardized based on the mean and standard deviation of each variable in the historical data under normal operating conditions, and the same mean and standard deviation are used to process online samples during subsequent online monitoring. This ensures that variables from different sensors are at the same scale, avoiding the unreasonable dominance of certain variables with larger dimensions in model training.

[0041] After standardization, historical operating data is divided into multiple continuous and non-overlapping time periods according to sampling time, and a time period indicator variable is introduced for each observation sample. This time period indicator variable is used to represent the time period to which the sample belongs, thereby describing how the statistical characteristics of the industrial process change over time. In actual industrial processes, load fluctuations, raw material changes, equipment aging, or operating condition adjustments can all cause slow changes in the mean and variance of process variables. By dividing the data into time periods and using time period indicator variables, these normal, phased changes can be distinguished during modeling, avoiding the direct treatment of normal non-stationary trends as faults.

[0042] Then, a Robust Probabilistic Stationary Subspace Analysis (RPSSA) model is constructed by combining the time-period indicator variables. This model represents the observed samples as the sum of process components, process noise, and outlier vectors after applying a mixture matrix to the potential sources. The potential sources include stationary and non-stationary sources. Stationary sources describe relatively stable process relationships across time periods, while non-stationary sources describe normal trends that change over time. Specifically, the mean and covariance of stationary sources are shared across time periods, while the mean and covariance of non-stationary sources vary over time. This configuration prevents the model from simply classifying all changes as outliers; instead, it expresses stable process relationships and normal non-stationary trends separately, which helps reduce false alarms caused by drift from normal operating conditions.

[0043] Furthermore, in the robust probabilistic stationary subspace analysis model, the Student's t-distribution is used for heavy-tailed modeling of potential sources. Compared to the ordinary Gaussian distribution, the Student's t-distribution has a heavier tail and is less sensitive to a small number of anomalous observations with large amplitudes. Transient interferences, brief measurement spikes, or occasional sensor anomalies commonly found in actual industrial data usually manifest as sparse but large-amplitude outliers. If modeled using the Gaussian distribution, these samples easily skew the estimation of the mean, covariance, and mixture matrix. By using the Student's t-distribution, the influence of outliers on model parameters is weakened, thus making the normal operating condition model closer to the actual normal operating state.

[0044] In its implementation, the Student's t-distribution is represented as a mixture of Gaussian and gamma distributions, and a scaling variable is introduced. This scaling variable participates in subsequent posterior inference; when an observed sample significantly deviates from the potential source distribution of its time period, the corresponding scaling variable reduces the sample's influence weight in parameter updates. Thus, outliers are not simply deleted, but participate in training within the probabilistic model with lower weights, enabling the model to balance data utilization and robustness. Simultaneously, time period indicator variables, mixed weights, a Dirichlet prior of the mixed weights, and a gamma prior of degrees of freedom are introduced to describe the uncertainties of sample time period attribution, the proportion of samples in each time period, and the tail thickness of the Student's t-distribution. This forms a robust probabilistic model for jointly describing stationary sources, non-stationary sources, process noise, and outlier interference.

[0045] After constructing the robust probabilistic model, the variational Bayesian expectation-maximization algorithm is used to train the model. During training, the time period indicator variable, latent sources, scale variable, degrees of freedom, and mixture weights are used as latent variables for posterior inference, and the mixture matrix, process noise variance, stationary source mean, stationary source covariance, non-stationary source mean, and non-stationary source covariance are updated simultaneously. Since there is a coupling relationship between time period attribution, outlier weights, latent source estimation, and model parameters, independent estimation can easily lead to error propagation. This embodiment alternates between the variational Bayesian expectation step and the variational Bayesian maximization step, enabling the posterior distribution of latent variables and model parameters to be updated collaboratively within the same probabilistic framework, which helps improve the stability of the training process.

[0046] In the variational Bayes expectation step, the posterior distributions of each latent variable are updated based on the current model parameters. These include the posterior distributions of the time period indicator variable, the latent source, the scale variable, the degrees of freedom, and the mixture weights. The posterior update of the scale variable is related to the degree to which a sample deviates from the latent source distribution of its respective time period, thus automatically adjusting for the influence of outliers during iteration. Subsequently, in the variational Bayes maximization step, the mixture matrix, process noise variance, and the mean and covariance of stationary and non-stationary sources are updated based on the posterior expectations of the latent variables. After multiple iterations, a trained model representing normal operating conditions is obtained.

[0047] After obtaining the trained model, the stationary source statistics and Wasserstein distance statistics are calculated using the training samples, and the corresponding control limits are determined. The stationary source statistics measure the deviation of the estimated stationary source values ​​from the normal distribution of stationary sources. This statistic changes when stable coupling relationships or long-term equilibrium relationships in an industrial process are disrupted, thus it can be used to monitor anomalies in stationary components. The Wasserstein distance statistic is constructed based on process uncertainty estimates. Specifically, process uncertainty estimates are obtained based on the differences between the observed samples and the process components reconstructed from potential sources. Then, a sliding window is used to collect process uncertainty estimates at multiple consecutive sampling times, and their empirical mean and empirical covariance are calculated. Based on these, the Wasserstein distance statistic is calculated. This statistic considers not only the mean shift of process uncertainty but also changes in covariance, thus reflecting changes in process noise, disturbance levels, or sensor accuracy. By combining the stationary source statistics and the Wasserstein distance statistic, the operating status can be evaluated from both the perspectives of stable process relationships and the distribution of process uncertainty, reducing the risk that a single statistic may not be sufficient to cover different fault manifestations.

[0048] Control limits can be determined based on the distribution of statistics obtained from the training samples. For example, the kernel density estimation method can be used to determine the control limits for stationary source statistics and Wasserstein distance statistics under a preset confidence level. Since the control limits are derived from training samples under normal operating conditions, they can reflect the fluctuation range of statistics under normal conditions, facilitating unified judgment during subsequent online monitoring.

[0049] During online monitoring, online operating samples of the industrial process control system are acquired and standardized using standardized parameters saved during the offline phase. The standardized online operating samples are then input into the training model, and latent variable inference is performed based on the trained model parameters to obtain the latent source estimates and process uncertainty estimates corresponding to the online samples. Then, the online stationary source statistics and online Wasserstein distance statistics are calculated separately. If either the online stationary source statistic or the online Wasserstein distance statistic exceeds its control limit, a fault is determined in the industrial process, and an alarm is output; if neither statistic exceeds its corresponding control limit, the industrial process is determined to be in normal operating condition.

[0050] In this online monitoring process, the training model is primarily based on historical data from normal operating conditions. Online samples are used to calculate statistics and determine status, rather than continuously incorporating online samples into the normal model for recursive updates. This avoids the model gradually adapting to fault samples after a fault occurs, reducing the possibility of fault detection delays. For scenarios requiring further fault location, the contribution of each process variable to the stationary source statistic or Wasserstein distance statistic can be calculated after an alarm is triggered to aid in identifying potentially abnormal variables or sensors.

[0051] In a preferred embodiment, after dividing the multivariate historical operating data under normal operating conditions into multiple consecutive and non-overlapping time periods, a time period indicator variable is set for the t-th observation sample to characterize the time period to which the observation sample belongs. The time period indicator variable satisfies: ; in, Indicates whether the t-th observation sample belongs to the i-th time period; This indicates that the t-th observation sample belongs to the i-th time period; This indicates that the t-th observation sample does not belong to the i-th time period; i represents the time period number; t represents the sampling time or sample number of the observation sample; n represents the number of consecutive and non-overlapping time periods.

[0052] The aforementioned constraints ensure that each observed sample corresponds to only one time period in a single modeling iteration, preventing the same sample from being assigned to multiple time periods and causing overlapping statistical parameters. For non-stationary industrial processes, statistical characteristics such as mean and covariance may differ across time periods due to load fluctuations, operating condition adjustments, or slow changes in equipment status. By setting the aforementioned time period indicator variables, the correspondence between samples and time periods can be clearly defined in subsequent robust probabilistic stationary subspace analysis models. This allows stationary source parameters to be shared across time periods, while non-stationary source parameters can be described separately for each time period, thus facilitating the differentiation between normal phased changes and abnormal deviations.

[0053] During model training, the time-period indicator variables and mixed weights jointly participate in the posterior inference of latent sources and the updating of time-period parameters. In this way, the model can estimate the mean and covariance of non-stationary sources based on the statistical differences of samples within different time periods, while maintaining the consistency of the mean and covariance of stationary sources across different time periods. This approach avoids directly treating normal non-stationary trends as faults, but rather represents them piecewise within the model, thus reducing false alarms caused by drift in normal operating conditions.

[0054] As a preferred embodiment, after standardizing and dividing the multivariate historical operating data into time periods, a robust probabilistic stationary subspace analysis model is constructed. This model is used to describe the relationship between observed samples, potential sources, process noise, and outlier disturbances in a non-stationary industrial process containing outliers. For the standardized observed samples at the t-th sampling time, the model satisfies: ; ; In the formula, x(t) is the standardized observation sample at the t-th sampling time; Let A represent the D-dimensional real space; D is the dimension of the process variables of the observed samples; N is the total number of samples; A is the invertible mixing matrix; s(t) is the potential source vector; e(t) is the process noise; and o(t) is the outlier vector.

[0055] In the above model, the mixing matrix A is represented as: ; Among them, each submatrix satisfies: ; In the formula, Let be a submatrix of a Spanning stationary subspace. Let be a submatrix spanning a nonstationary subspace; d is the dimension of the stationary source, and Dd is the dimension of the nonstationary source. By decomposing the mixture matrix into submatrices corresponding to the stationary and nonstationary subspaces, the subsequent model can describe the stable process relationships and normal nonstationary changes respectively, avoiding treating all statistical changes as outliers.

[0056] The potential source vector s(t) is represented as: ; Among them, each potential source satisfies: ; In the formula, For a stable source, For non-stationary sources, T represents the vector transpose. Stationary sources are used to characterize relatively stable process relationships in each time period, while non-stationary sources are used to characterize normal trends that change over time. The dimension d of the stationary source can be determined through the Johansen cointegration test. Through the above source space partitioning, normal non-stationary trends and stable process relationships are represented separately, which helps to reduce false alarms caused by drift in normal operating conditions.

[0057] In one specific implementation, process noise It follows a Gaussian distribution with zero mean: ; In the formula, N(·) represents a Gaussian distribution; Indicates the process noise variance; This represents a D-dimensional identity matrix. Process noise is used to describe random fluctuations during normal operation that are difficult to fully explain by potential sources. Representing it separately from outlier vectors helps avoid confusing general process noise with outlier disturbances.

[0058] The outlier vector o(t) represents the impact of outliers at the observation level, such as transient disturbances, measurement spikes, or sporadic sensor anomalies. Since outliers are typically few in number but can have large amplitudes, directly estimating model parameters using the ordinary Gaussian assumption can easily lead to biases in the estimation of the mixture matrix, mean, and covariance. This embodiment explicitly preserves the outlier vector in the observation model and subsequently performs robust modeling of potential sources using the Student's t-distribution and its scaling variable, thus suppressing the influence of samples deviating from their respective time-period distributions in parameter updates. This approach both reflects the existence of outliers at the observation level and reduces the impact of a small number of abnormal samples on the normal operating condition model.

[0059] Specifically, such as Figure 2 As shown, each observed sample x(t) in the industrial process is considered as an observation formed by mapping the potential source s(t) through the mixing matrix A and superimposing process noise. The potential source s(t) includes stationary and non-stationary sources. For the t-th sample, a time period indicator variable r(t) is introduced to indicate which of several consecutive non-overlapping time periods the sample belongs to. The mixing weight π is used to characterize the probability of each time period being selected, and its prior is determined by the parameter β0. By combining r(t) and π, the phased statistical changes of the industrial process can be included in the model description, avoiding the direct judgment of normal operation switching or slow drift as a fault.

[0060] Figure 2 μ in i and Σ i Let be the mean and covariance of the potential sources corresponding to the i-th time period, respectively, used to describe the statistical distribution of the potential sources within that time period. According to the modeling method of this invention, the mean and covariance of stationary sources are shared across time periods, while the mean and covariance of non-stationary sources change with time. Therefore, the model can simultaneously express stable process relationships and normal non-stationary trends.

[0061] Figure 2 The scaling variable λ(t) and the degrees of freedom ν are used together to model the heavy-tailed distribution of the Student's t-distribution. The prior of the degrees of freedom ν is given by a. ν and b ν The scaling variable λ(t) is determined to moderate the influence of the t-th sample in parameter estimation. When an observed sample deviates significantly from the potential source distribution of its time period, the scaling variable can reduce the influence of that sample on model parameter updates, thereby mitigating the interference of outliers on the estimation of the mixing matrix, mean, and covariance.

[0062] Figure 2 In this model, A and σ² correspond to the mixture matrix and process noise variance, respectively. The mixture matrix A is used to establish the mapping relationship between potential sources and observed samples, while the process noise variance σ² is used to describe random disturbances not explained by potential sources during the observation process. Based on the above model, the mean and covariance of stationary sources can be shared across different time periods, while the mean and covariance of non-stationary sources can vary with time. The model parameters can then be solved using the variational Bayesian expectation-maximization algorithm. This modeling approach can describe normal non-stationary trends, process noise, and outlier disturbances within the same framework, providing a model foundation for the subsequent construction of stationary source statistics and process uncertainty distribution statistics.

[0063] In a preferred embodiment, to distinguish between stable process relationships and normal time-varying trends in non-stationary industrial processes, potential sources are classified into stationary and non-stationary sources. For the resulting classification... For each time period, the mean and covariance of the potential sources are set as follows: ; In the formula, Indicates the first The mean of potential sources over a period of time; Indicates the first Potential source covariance for each time period; This represents the mean of a stationary source; Indicates the covariance of stationary sources; Indicates the first The mean of non-stationary sources over a period of time; Indicates the first The non-stationary source covariance for each time period; the zero elements in the matrix represent that there is no correlation between the stationary and non-stationary source groups.

[0064] With the above settings, the mean and covariance of the stationary source are shared across time periods, used to describe the relatively stable process relationships under normal operating conditions; the mean and covariance of the non-stationary source vary with time periods, used to describe normal time-varying trends caused by load fluctuations, raw material changes, or operating condition adjustments. Therefore, when performing subsequent fault detection, the model does not need to treat all statistical changes as anomalies, which helps reduce false alarms caused by normal non-stationary changes.

[0065] Furthermore, to reduce the impact of outliers on model parameter estimation, a Student's t-distribution is used for heavy-tailed modeling of the potential sources. In the... Within a given time period, the probability density of the potential source satisfies: ; In the formula, Indicates the first The probability density of potential sources within a given time period; Indicates the first The observation sample at the ... Potential sources corresponding to each time period; Represent the t-distribution of students; Indicates the first The mean of potential sources over a period of time; Indicates the first Potential source covariance for each time period; Indicates degrees of freedom.

[0066] The student t-distribution can be represented as a mixture of Gaussian and gamma distributions: ; In the formula, Indicates a Gaussian distribution; Indicates the gamma distribution; Represents a scale variable; This represents the covariance adjusted for the scaling variable; the meanings of the other symbols are the same as described above.

[0067] Under the above representation, the scaling variable is related to the degree to which a sample deviates from the potential source distribution of its time period. When an observed sample deviates from the normal distribution due to transient disturbances, measurement spikes, or occasional sensor anomalies, the scaling variable can reduce the influence weight of that sample in subsequent parameter updates. Therefore, the model does not need to pre-emptively remove suspected outliers and can also mitigate the biasing effect of a small number of outlier samples on the estimates of the mixture matrix, mean, and covariance.

[0068] Furthermore, to describe the proportional relationship of different time periods in the training data, a Dirichlet prior is introduced for the mixed weights: ; In the formula, Represents the mixed weight vector; Indicates that the observed sample belongs to the first Prior weights for each time period; Indicates the Dirichlet distribution; Represents the Dirichlet prior parameter vector; Indicates the relationship with the first Prior parameters corresponding to each time period; The normalization constant of the Dirichlet distribution is represented by . Indicates the number of time periods; the multiplication symbol in the formula indicates multiplication over all... Calculate the product of the corresponding items for each time period; This represents the transpose of a vector.

[0069] By setting a Dirichlet prior for the mixed weights, a prior constraint on the proportion of samples in each time period can be introduced into the time period attribution inference, avoiding the excessive influence of individual samples or a small number of outliers on the time period weight estimation, thereby making the probability modeling under time period division more stable.

[0070] Furthermore, a gamma prior is introduced for the degrees of freedom of the student's t-distribution: ; In the formula, Degrees of freedom The prior probability density; This represents the degrees of freedom of the student's t-distribution; Indicates the gamma distribution; and These represent the prior shape parameter and rate parameter for the degrees of freedom, respectively.

[0071] Degrees of freedom are used to characterize the tail thickness of the Student's t-distribution. By setting a gamma prior for the degrees of freedom, the degree of heavy tails can be probabilistically constrained during model training, enabling the model to adjust its robustness based on the influence of outliers in the data, rather than relying on a fixed threshold to identify outliers. This is beneficial for improving the stability of modeling non-stationary industrial processes containing outliers and provides a more reliable normal-condition model basis for subsequent fault detection statistics calculations.

[0072] In a preferred embodiment, after constructing the robust probabilistic model, this embodiment employs the Variational Bayesian Expectation-Maximization (VBEM) algorithm to solve for the model parameters. Since the latent sources use a Student's t-distribution for heavy-tailed modeling, and the model simultaneously includes latent variables such as time-period indicator variables, scale parameters, degrees of freedom, and mixed weights, the joint posterior distribution of each latent variable is difficult to obtain directly analytically. Using the VBEM algorithm allows for alternating updates of the latent variable posterior distribution and deterministic model parameters within the same probabilistic framework. Specifically, the VBEM algorithm alternately executes the Variational Bayesian Expectation step (VBE step) and the Variational Bayesian Maximization step (VBM step), where the VBE step updates the posterior distribution of the latent variable set, and the VBM step updates the deterministic parameter set. This setup avoids error propagation caused by independently estimating time-period attribution, outlier weights, and model parameters.

[0073] In this embodiment, the unknown variables are divided into a set of random variables and a set of deterministic parameters. The set of random variables is represented as follows: ; The deterministic parameter set is represented as: ; In the formula, Represents a set of random variables. Indicates the first The time period indicator variable for each sample, Indicates the first The potential source corresponding to each sample Indicates the scale parameter. This represents the degrees of freedom of the student's t-distribution. Indicates mixed weights; Represents a deterministic set of parameters. This represents the mean of a stationary source. This represents the mean of a non-stationary source. Indicates the covariance of stationary sources. This represents the covariance of a non-stationary source. Represents a mixture matrix. This represents the variance of process noise. The random variable set approximates its posterior distribution through variational inference, while the deterministic parameter set is updated by maximizing the expected log-likelihood of the complete data. This partitioning allows the influence of outliers, sample time period attribution, and latent source estimation to be determined collaboratively during training.

[0074] For observation data Its logarithmic marginal likelihood can be decomposed into: ; In the formula, This represents historical observation data under normal operating conditions. Indicates known prior parameters. This represents the variational posterior distribution of the latent variable. This represents the corresponding true posterior distribution. This represents the Kullback-Leibler divergence between the variational posterior distribution and the true posterior distribution. Indicates the lower bound of evidence. Because... Since the term is non-negative, maximizing the lower bound of evidence is equivalent to reducing the difference between the variational posterior and the true posterior, thereby obtaining an approximate optimal solution for the model.

[0075] To reduce the difficulty of solving the joint posterior distribution, this embodiment adopts the mean-field assumption, decomposing the variational posterior into a product of the posteriors of each variable group: ; In the formula, Represents a set of random variables The first in Group of variables, Let represent the variational posterior distribution of this variable set. Based on this decomposition, each variable set can be optimized sequentially until the lower bound of evidence is maximized. The optimal variational posterior of the set of variables satisfies: ; In the formula, Indicates the first The optimal variational posterior distribution of the groups of variables Indicates that except for the first Calculate the expectation of the variational posterior distribution of the variable groups other than the initial variable groups. This represents the joint probability distribution of observed data and latent variables under given deterministic and prior parameters. Indicates and Irrelevant constant terms. Thus, the complex joint posterior problem can be transformed into multiple alternately updated posterior problems, which facilitates stable model training on non-stationary industrial process data containing outliers.

[0076] Given a deterministic set of parameters In this case, the Variational Bayesian Expectation step is first executed to update the optimal variational posterior distribution of each latent variable.

[0077] Time period indicator variable The posterior distribution is: ; Among them, intermediate variables Defined as: ; In the formula, This represents the optimal variational posterior distribution of the time-period indicator variable. Indicates the first Does the _ sample belong to the _ ... Each period, Indicates the first The sample belongs to the first The posterior probability for each time period Indicates calculation The intermediate quantity, Indicates the dimension of the process variable. Indicates the scale parameter. Indicates the first Potential source covariance over time period Indicates a potential source. Indicates the first The mean of potential sources over a given period, Indicates the first The mixed weights corresponding to each time period This represents the expectation under the corresponding variational distribution. By updating... The model can determine the time period a sample belongs to based on the degree of matching between the sample and the potential source distribution of each time period, thus avoiding the rigid constraints imposed on boundary samples by fixed time period divisions.

[0078] Potential source The posterior distribution follows a Gaussian distribution: ; Wherein, the posterior covariance and the posterior mean are respectively: ; ; In the formula, Let represent the optimal variational posterior distribution of the potential source. Indicates a Gaussian distribution. This represents the posterior mean of the potential source. This represents the potential source posterior covariance. Represents a mixture matrix. Represents the process noise variance. Indicates the first One observation sample, , , and The meaning is the same as above. In the above update formula, the scale parameter and time period indicator variables They jointly influence the posterior estimation of potential sources, enabling outlier samples and time-series differences to be reflected in the estimation of potential sources.

[0079] Scale parameters The posterior follows a gamma distribution: ; Its parameters are: ; ; In the formula, This represents the optimal variational posterior distribution of the scaling parameter. Represents the gamma distribution. Indicates shape parameters, Representational rate parameter, This represents the degrees of freedom of the student's t-distribution. When a sample significantly deviates from the latent source distribution of its time period, the quadratic term in the formula increases, and the posterior scaling parameter is adjusted accordingly, thereby reducing the impact of the sample on the model parameter update. This approach can suppress the bias of outliers on the model without directly deleting samples.

[0080] Degrees of freedom The posterior follows a gamma distribution: ; Its parameters are: ; ; In the formula, The optimal variational posterior distribution representing the degrees of freedom. and These represent the updated gamma distribution parameters, and These represent the parameters of the gamma prior for the degrees of freedom. This represents the number of training samples. The degrees of freedom reflect the tail thickness of the student's t-distribution. By updating the degrees of freedom posteriorly, the model can adaptively adjust its heavy-tailed characteristics based on the degree of outliers in the training data.

[0081] Mixed weights The posterior follows a Dirichlet distribution: ; The updated parameters are: ; In the formula, Describes the optimal variational posterior distribution of the mixed weights. Indicates the Dirichlet distribution. Represents the mixed weight vector. This indicates the updated Dirichlet parameters. Indicates the first Update parameters corresponding to each time period Indicates the first The prior parameters correspond to each time period. This update method can adjust the mixing weights based on the posterior attribution of samples in each time period, making the time period proportion estimation more consistent with the actual data distribution.

[0082] After fixing the posterior distribution of the latent variables, a variational Bayesian maximization step is performed to update the deterministic parameters by maximizing the expected log-likelihood of the complete data. Among them, the hybrid matrix The update formula is: ; In the formula, Represents a mixture matrix. Indicates the first training samples, Indicates a potential source. Indicates the number of training samples. This represents the expectation of the potential source transpose. This represents the expectation of the second moment of the latent source. This update uses the posterior information of the latent sources from all training samples to estimate the mapping relationship between process variables and latent sources, which helps to obtain a stable representation of process components.

[0083] Process noise variance The update formula is: ; In the formula, Represents the process noise variance. This indicates the dimension of the process variable; the other symbols have the same meaning as above. Using this update formula, the process noise level under normal operating conditions can be estimated based on the deviation between the observed samples and the reconstructed results of the potential sources.

[0084] stationary source mean The update formula is: ; Non-stationary source mean The update formula is: ; In the formula, This represents the mean of a stationary source. Indicates the first The mean of non-stationary sources over a given period This represents the stationary source selection matrix. This represents the non-stationary source selection matrix. The mean of stationary sources is shared across all time periods, while the mean of non-stationary sources is updated separately for each time period. This approach preserves the stable process relationships under normal operating conditions while also describing the normal trend changes at different time stages.

[0085] make: ; Then the stationary source covariance The update formula is: ; Non-stationary source covariance The update formula is: ; In the formula, This represents the residual vector of a stationary source deviating from its mean. Indicates the first The residual vector of a non-stationary source deviating from its mean over a given time period. Indicates the covariance of stationary sources. Indicates the first The covariance of non-stationary sources over a given time period is updated using the covariance method described above. This allows for a unified estimation of the fluctuation structure of stationary sources across all time periods, while the fluctuation structure of non-stationary sources is estimated separately for each time period, thereby reducing the interference of normal non-stationary changes on the fault detection model.

[0086] Lower bound sequence of evidence generated by the VBEM algorithm The expression is monotonically non-decreasing and bounded above; therefore, the iterative process converges to a finite limit. The termination condition for the iteration is: ; In the formula, Indicates the first The lower bound of evidence in the next iteration Indicates the first The lower bound of evidence in the next iteration This represents the convergence threshold, which can typically be set to a value that is not explicitly stated in the original text. By setting this convergence condition, iteration can be stopped when the change in the lower bound of evidence tends to stabilize, avoiding unnecessary calculations and ensuring that the parameter estimation results have good stability.

[0087] As a preferred embodiment, after completing the training of the robust probabilistic stationary subspace analysis model and obtaining the training model, the potential source and process uncertainty of the sample are estimated using the training model, and stationary source statistics and Wasserstein distance statistics are constructed accordingly to simultaneously monitor the deviation of stationary components and changes in the distribution of process uncertainty, so as to avoid the problem of insensitivity to some fault performance when relying on only a single statistic.

[0088] Specifically, for the first For each sample, the potential source estimate is first obtained based on the trained model. And select the matrix through the stationary source. Extract the stationary source estimates: ; Furthermore, based on the mean and covariance of the latent source estimates obtained during the training phase, the stationary source statistic is calculated: ; In the formula, Indicates the first Stationary source statistics for each sample; Indicates the first Potential source estimates for each sample; Indicates the first Stationary source estimates for each sample; This represents the selection matrix used to extract stationary sources from potential sources; This represents the mean of the potential source estimates during the training phase; Indicates the covariance of the latent source estimates during the training phase; superscript This represents the transpose of a vector.

[0089] The aforementioned stationary source statistic essentially measures the degree of deviation of the stationary source estimate from the normal training distribution. Since stationary sources represent relatively stable process relationships over different time periods, this statistic increases when the variable coupling relationships or stable operating structure of an industrial process become abnormal. Therefore, this statistic can be used to monitor anomalies in stationary components and helps distinguish between normal non-stationary trends and deviations from the true process state.

[0090] To further monitor changes in process noise and disturbance distribution, this embodiment also calculates process uncertainty estimates based on the differences between the observed samples and the process components reconstructed from potential sources: ; In the formula, Indicates the first Estimated process uncertainty for a single sample; Indicates the first Observations of a sample; Represents a mixture matrix; Indicates the first Potential source estimates for each sample.

[0091] Considering that the process uncertainty at a single sampling moment may be affected by random fluctuations, this embodiment uses a length of... The sliding window collects process uncertainty estimates from multiple consecutive sampling times: ; In the formula, Indicates as of the date A set of process uncertainty estimates within a sliding window for each sample; Indicates the length of the sliding window; to These represent the estimated process uncertainty values ​​at each sampling time point within the sliding window.

[0092] Calculate the empirical mean of the set of process uncertainty estimates within the sliding window. and experience covariance The Wasserstein distance statistic is calculated according to the following formula: ; In the formula, Indicates the first Wasserstein distance statistic for each sample; This represents the empirical mean of the process uncertainty estimates within the sliding window. The empirical covariance representing the estimate of process uncertainty within the sliding window; Indicates the dimension of the process variable; Indicates the standard deviation of process noise; Represents the trace of a matrix; Represents the L2 norm; Representing the empirical covariance matrix The square root of the matrix.

[0093] The Wasserstein distance statistic measures the deviation of the empirical distribution of process uncertainty from the normal noise distribution within a sliding window. Compared to statistics that only consider residual amplitudes, this statistic considers changes in both the empirical mean and empirical covariance, thus reflecting variations in process disturbance levels, noise structure, or sensor accuracy. For anomalies that do not immediately disrupt the stationary source structure but have already caused changes in the process uncertainty distribution, this statistic can provide supplementary monitoring information.

[0094] During the offline training phase, calculations are performed using the training samples respectively. and The control limits for the stationary source statistic and the Wasserstein distance statistic are determined based on the distribution of the training sample statistics. During the online monitoring phase, the above calculation process is repeated for the online samples. If the online stationary source statistic exceeds its control limit, or the online Wasserstein distance statistic exceeds its control limit, the industrial process is deemed to have malfunctioned and an alarm message is output. If neither statistic exceeds its corresponding control limit, the industrial process is deemed to be in normal operation.

[0095] With the above settings, the stationary source statistic focuses on monitoring deviations from stable relationships in the stationary subspace, while the Wasserstein distance statistic focuses on monitoring changes in the distribution of process uncertainty. Combining the two allows for evaluation of the operating status from both process structure changes and noise distribution changes, making the fault detection method of this application more applicable to industrial process data scenarios where outliers and non-stationary trends coexist.

[0096] In a preferred embodiment, after training the robust probabilistic stationary subspace analysis model, the stationary source statistic and Wasserstein distance statistic obtained from the training samples are calculated, and kernel density estimation is performed at a preset confidence level. Determine the control limits for stationary source statistics Control limits for Wasserstein distance statistic Among them, the stationary source statistic reflects the degree of deviation of the stable process relationship in the stationary subspace, and the Wasserstein distance statistic reflects the degree of deviation of the distribution of process uncertainty estimates relative to the normal state. Both control limits are obtained from training samples under normal operating conditions and can serve as the basis for judging whether the operating state deviates from the normal fluctuation range during the online monitoring phase.

[0097] Specifically, after obtaining the stationary source statistic and Wasserstein distance statistic for each training sample during the training phase, kernel density estimation is performed on the sample distributions of the two types of statistics, and a pre-set confidence level is established. The corresponding quantiles are then determined as control limits for stationary source statistics. Control limits for Wasserstein distance statistic Determining control limits in this way does not require prior assumption that the statistic strictly follows a specific analytical distribution. It can adapt to situations where the distribution of the training sample statistic has certain skewness or non-Gaussian characteristics, thus making the control limits closer to the actual range of variation of the statistic under normal operating conditions.

[0098] During online monitoring, the mixture matrix, process noise variance, mean of stationary sources, covariance of stationary sources, mean of non-stationary sources, and covariance of non-stationary sources in the fixed training model are used to infer only the time-period indicator variables, latent sources, and scale variables corresponding to the online operating samples. The purpose of this approach is to ensure that online samples are used only for state determination and latent variable estimation, without directly altering the model parameters trained from historical data under normal operating conditions. This reduces the risk of fault samples continuously participating in model updates and causing the model to gradually adapt to fault states, thus helping to maintain consistency between control limits and the trained model.

[0099] After obtaining the stationary source statistics and Wasserstein distance statistics corresponding to the online running samples, fault determination is performed according to the following rules: ; In the formula, Indicates the sampling time of the online running sample; Indicates the preset credit level; Indicates the control limits for stationary source statistics; Indicates the control limits of the Wasserstein distance statistic; Indicates the first Stationary source statistics for a single online running sample; Indicates the first Wasserstein distance statistics for each online running sample.

[0100] In the above judgment rules, when Not exceeding and Not exceeding When the deviation of the stationary components and the change in the distribution of process uncertainty in the online sample are within the allowable range of normal operating conditions, it is judged as normal; when Exceed or Exceed If this occurs, it indicates that the stable relationship in the stationary subspace has deviated abnormally, or that the distribution of the process uncertainty estimate has changed abnormally. Therefore, a fault is determined to have occurred and an alarm message is output.

[0101] This judgment method incorporates both stationary source anomalies and changes in process uncertainty distribution into the monitoring scope. For faults primarily manifested as disruptions in stable variable relationships, stationary source statistics provide corresponding detection criteria; for anomalies primarily manifested as changes in disturbance levels, noise distribution, or sensor accuracy, Wasserstein distance statistics provide supplementary judgment. The two types of statistics are used in conjunction with an alarm triggered by either exceeding a certain limit. This reduces the risk of missed detections due to insufficient coverage of fault types by a single statistic, while maintaining fixed training model parameters during the online phase helps maintain the stability of the model under normal operating conditions.

[0102] To further illustrate the inventive concept of this invention, the method provided by this invention is applied to fault detection in the industrial fluidized bed roasting process of a zinc smelter. For example... Figure 3 As shown, the industrial roasting process includes a feeding unit 1, a blower unit 2, a roasting furnace 3, a waste gas unit 4, and a roasting unit 5. The feeding unit 1 feeds zinc concentrate into the roasting furnace 3, the blower unit 2 provides the necessary air for roasting into the furnace 3, the roasted gas enters the waste gas unit 4 for cooling and treatment, and the roasted product is output through the roasting unit 5. The roasting process is the first step in zinc smelting, and the feed rate, blower volume, furnace conditions, and waste gas treatment process all affect the roasting operation. Because the blower volume and feed rate fluctuate in actual production, the statistical characteristics of the roasting process change over time; simultaneously, outliers may appear in the collected data due to data transmission, sampling disturbances, or occasional sensor anomalies. Therefore, this type of process is a typical non-stationary industrial process containing outliers, suitable for monitoring using the robust fault detection method described in this invention.

[0103] In practice, multiple process variables reflecting the operating status of the roasting process are first sampled from the industrial process control system. These process variables may include feed-related variables, blast-related variables, furnace operation variables, cooling-related variables, exhaust gas-related variables, and motor operating status variables. After standardizing the historical operating data collected under normal operating conditions, it is divided into multiple continuous and non-overlapping time periods, and time period indicator variables are introduced to describe the statistical changes in the roasting process at different operating stages. Through this processing, non-stationary changes caused by input fluctuations such as feed rate and blast rate under normal operating conditions can be included in the model description range, reducing the possibility of misjudging normal operating condition changes as faults.

[0104] In the offline modeling phase, a robust probabilistic stationary subspace analysis model is constructed based on standardized normal samples. The model represents the observed samples as the sum of process components, process noise, and outlier vectors after the mixing matrix is ​​applied to the potential sources, and classifies the potential sources into stationary and non-stationary sources. Among them, stationary sources are used to describe the relatively stable variable coupling relationships in different stages of the roasting process, while non-stationary sources are used to describe the normal trends that change with the operation stage.

[0105] Furthermore, a student's t-distribution is used to model the potential sources with heavy tails, and a scaling variable is introduced through a mixture of Gaussian and gamma distributions. For outliers caused by unstable data transmission, transient disturbances, or measurement spikes, the scaling variable can reduce the influence weight of such samples during parameter updates, making the mixing matrix, noise variance, and parameters of stationary and non-stationary sources less susceptible to bias by a small number of abnormal samples. Therefore, the trained normal operating condition model can better reflect the normal operating rules of the roasting process.

[0106] In a specific test, 10 variables during the roasting process were sampled, and 800 normal state samples were collected for modeling. 400 samples were collected before and after the fault for testing. The number of stationary sources was determined using the Johansen test, and the number of time periods was determined heuristically. The model iteration error, sliding window width, and control limit confidence level could be set according to the sample size and on-site monitoring requirements. In this embodiment, the control limit was determined using kernel density estimation at a confidence level of 0.99, and the sliding window width of the Wasserstein distance statistic was set to 3.

[0107] During online monitoring, online operational samples of the roasting process are input into the training model to calculate online stationary source statistics and online Wasserstein distance statistics. The stationary source statistics are used to determine whether the relationships between stable variables have deviated during the roasting process; the Wasserstein distance statistics are determined based on the empirical mean and empirical covariance of the sliding window of process uncertainty estimates, and are used to determine whether there have been distribution shifts in process noise, disturbance levels, or sensor states. When either statistic exceeds its corresponding control limit, an anomaly is detected in the roasting process, and an alarm is output.

[0108] like Figure 4 As shown, different methods yield different monitoring results for the same roasting process. Specifically, Figure 4 (a) and Figure 4(b) shows the monitoring results of the RPCA (Recursive Principal Component Analysis) method, in which the T² statistic and SPE statistic respond to some anomalous changes. However, this type of method may gradually adapt to fault samples due to online updates, and there is a risk of delay or weakening of fault detection results. Figure 4 (c) shows the T² statistic monitoring results of the SSA (Stationary Subspace Analysis) method, which does not respond significantly to fault changes in this embodiment. Figure 4 (d) shows the T² statistic monitoring results of the EASSA (Exponential Analytic Stationary Subspace Analysis) method. The statistic shows a certain over-limit response during the fault stage, but its false alarm rate is relatively high, indicating that its detection sensitivity may be affected by the control limit setting. Figure 4 (e) and Figure 4 (f) shows the PSSA (Probabilistic Stationary Subspace Analysis) method. Statistics and The statistical monitoring results have limited ability to identify faults in this embodiment. Figure 4 (g) and Figure 4 Figure (h) shows the monitoring results of the RPSSA (Robust Probabilistic Stationary Subspace Analysis) method of this invention. Figure 4 (g) is shown Statistics are mainly used to reflect deviations from stationary sources; Figure 4 The (h) shown WD e The statistic is mainly used to reflect the distribution change of process uncertainty estimates within a sliding window. In this embodiment, WD e The fact that the statistics consistently exceed limits after the fault occurs indicates that the process uncertainty distribution has deviated from the normal state. In contrast, in the robust probabilistic stationary subspace analysis model used in this invention, the stationary source statistics and Wasserstein distance statistics reflect the process state from different perspectives, with the Wasserstein distance statistic showing a more significant response to changes in the process uncertainty distribution under this fault scenario.

[0109] In such Figure 3In the industrial roasting process illustrated, input fluctuations, operating condition switching, and data disturbances collectively lead to non-stationarity and outlier interference in the process data. This invention describes the normal operating trend through stationary / non-stationary source decomposition, reduces the impact of outliers on parameter estimation through Student's t-tailed modeling, and uses joint monitoring with stationary source statistics and Wasserstein distance statistics. This allows fault detection to focus not only on deviations from stable process relationships but also on changes in the distribution of process uncertainty. Therefore, this invention is applicable to online fault detection in such non-stationary industrial processes containing outliers.

[0110] Example 2

[0111] This invention also provides a robust fault detection system for outlier, non-stationary industrial processes. The system can be deployed in existing industrial process control systems to perform normal operating condition data modeling, online operational status monitoring, and fault alarms. The system includes industrial process sensors, a data processing server, and an alarm terminal, and these components can be connected via industrial Ethernet, fieldbus, wireless communication networks, or internal interfaces of the control system.

[0112] The industrial process sensors are used to collect multivariate historical operating data and online operating samples of industrial processes under normal operating conditions. The sensors may include detection devices related to process status, such as temperature, pressure, flow rate, material quantity, gas parameters, motor operating status, and equipment vibration. The collected data is organized into multivariate observation samples according to the sampling time and sent to a data processing server. Simultaneous multivariate acquisition provides a data foundation for subsequent decomposition of stationary and non-stationary sources, avoiding insufficient process status characterization due to reliance on a single variable.

[0113] The data processing server includes a processor, a memory, and a communication interface. The memory stores a program executable by the processor, which, when executed, implements the robust fault detection method described in Embodiment 1 of this invention. Specifically, the data processing server standardizes historical data under normal operating conditions and divides it into multiple continuous, non-overlapping time periods, introducing time period indicator variables to describe the phased changes in the statistical characteristics of the industrial process. This processing can incorporate normal non-stationary trends caused by load fluctuations, operating condition adjustments, or slow equipment changes into the model's description scope, reducing the possibility of normal changes being misjudged as faults.

[0114] During the offline modeling phase, the data processing server constructs a robust probabilistic stationary subspace analysis model based on standardized historical data. The observed samples are represented as the sum of process components, process noise, and outlier vectors after a mixing matrix is ​​applied to the potential sources. The potential sources are categorized into stationary and non-stationary sources. The mean and covariance of stationary sources are shared across time periods, while the mean and covariance of non-stationary sources vary with time. This structure allows the system to describe stable process relationships and normal non-stationary trends separately, reducing the interference of non-stationary trends in fault diagnosis.

[0115] Furthermore, the data processing server employs the Student's t-distribution for heavy-tailed modeling of potential sources, and introduces scaling variables, time-period indicator variables, mixed weights, and corresponding prior distributions through a mixture of Gaussian and gamma distributions to form a robust probabilistic model. For transient interferences, measurement spikes, or occasional sensor anomalies that may occur in industrial settings, the scaling variable can reduce the impact of samples deviating from their respective time-period distributions on parameter updates, making the normal operating condition model less susceptible to being skewed by a small number of outliers.

[0116] During the model training phase, the data processing server uses the variational Bayesian expectation-maximization algorithm to update the posterior distribution of latent variables, as well as the mixture matrix, process noise variance, and the mean and covariance of stationary and non-stationary sources, to obtain the trained model. This training process treats sample time period assignment, latent source estimation, outlier influence, and model parameter updates within the same probabilistic framework, which helps reduce error propagation caused by step-by-step estimation. After training is complete, the data processing server calculates the stationary source statistics and Wasserstein distance statistics based on the training samples and determines the corresponding control limits.

[0117] During online monitoring, industrial process sensors continuously or at preset intervals collect online operational samples and send them to a data processing server. The data processing server processes the online samples using standardized parameters saved during the offline modeling phase, and infers the time-period indicator variables, potential sources, and scale variables corresponding to the online samples under the condition of fixed training model parameters. Therefore, online samples are primarily used for state determination rather than being continuously incorporated into normal model updates, which reduces the risk of the model gradually adapting to the fault state after a failure occurs.

[0118] The data processing server calculates online stationary source statistics and online Wasserstein distance statistics based on the inference results of online samples. The stationary source statistics reflect deviations from stable relationships in the stationary subspace, while the Wasserstein distance statistics reflect the distribution changes of process uncertainty estimates within a sliding window. When either statistic exceeds its corresponding control limit, the data processing server outputs a fault alarm; when neither statistic exceeds its corresponding control limit, it outputs a normal status message.

[0119] The alarm terminal is used to receive status information output by the data processing server and display the normal or fault status of the industrial process. The alarm terminal can be a human-machine interface in the control room, a host computer monitoring interface, a mobile terminal, or an audible and visual alarm device. When a fault alarm occurs, the alarm terminal can display the type of out-of-limit statistical quantity, sampling time, corresponding control limit, and alarm prompt information, facilitating timely inspection and handling by on-site personnel.

[0120] In a preferred embodiment, the data processing server can also store alarm records, statistical curves, control limits, and online sample time information for subsequent traceability analysis. For scenarios requiring fault location, the contribution of each process variable to the stationary source statistic or Wasserstein distance statistic can also be calculated to help identify potentially abnormal variables or sensors.

[0121] Through the above structure, industrial process sensors are responsible for collecting process data, the data processing server is responsible for robust modeling, model training, online inference, and statistical calculation, and the alarm terminal is responsible for outputting identifiable operating status. This system can deploy robust fault detection methods for outlier non-stationary industrial processes in actual industrial control environments, providing a foundation for online fault detection in scenarios where non-stationary trends and outlier disturbances coexist.

[0122] Example 3

[0123] The present invention also provides an electronic device, including a memory for storing computer program instructions and a processor for executing the computer program instructions, wherein when the computer program instructions are executed by the processor, the device is triggered to execute some or all of the steps in Embodiment 1; A processor may include one or more processing units, such as an application processor (AP), a modem processor, a graphics processing unit (GPU), an image signal processor (ISP), a controller, a video codec, a digital signal processor (DSP), a baseband processor, and / or a neural network processing unit (NPU). Different processing units may be independent devices or integrated into one or more processors.

[0124] The controller can serve as the nerve center and command center of an electronic device. Based on the instruction opcode and timing signals, the controller generates operation control signals to control the fetching and execution of instructions.

[0125] The processor may also include memory for storing instructions and data. In some embodiments, the memory in the processor is a cache memory. This memory can store instructions or data that the processor has just used or that are used repeatedly. If the processor needs to use the instruction or data again, it can retrieve it directly from the memory. This avoids repeated accesses, reduces processor waiting time, and thus improves system efficiency.

[0126] Example 4

[0127] The present invention also provides a storage medium having a computer program stored thereon, wherein, when the program is executed, it controls the device where the storage medium is located to perform some or all of the steps in Embodiment 1.

[0128] The storage medium may include high-speed RAM memory, and may also include nonvolatile memory, such as at least one disk storage device. It is understood that the storage medium can be any machine-readable medium capable of storing program code, such as random access memory (RAM), magnetic disk, hard disk, solid state disk (SSD), or nonvolatile memory.

[0129] Those skilled in the art will understand that embodiments of the present invention can be provided as methods or storage media. Therefore, embodiments of the present invention can take the form of entirely hardware embodiments, entirely software embodiments, or embodiments combining software and hardware aspects. Furthermore, embodiments of the present invention can take the form of a computer program product implemented 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.

[0130] The above are merely preferred embodiments of the present invention and do not limit the patent scope of the present invention. Any equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct or indirect applications in other related technical fields, are within the patent protection scope of the present invention.

Claims

1. A robust fault detection method for outlier-oriented non-stationary industrial processes, characterized in that, Includes the following steps: The system acquires multivariate historical operating data collected by multiple sensors in an industrial process control system under normal operating conditions. After standardization, the data is divided into multiple continuous non-overlapping time periods, and time period indicator variables are introduced. A robust probabilistic stationary subspace analysis model is constructed by combining time-period indicator variables. The observed sample is represented as the sum of process components, process noise, and outlier vectors after the mixture matrix is ​​applied to the potential source. The potential source includes stationary and non-stationary sources. The mean and covariance of the stationary source are shared in each time period, while the mean and covariance of the non-stationary source change with time period. The student t-distribution is used to model the potential sources with heavy tails. A robust probability model is formed by introducing a scale variable, a time period indicator variable, a mixture weight, a Dirichlet prior of the mixture weight, and a gamma prior of the degrees of freedom through a mixture of Gaussian and gamma distributions. The robust probability model is trained using the variational Bayesian expectation-maximization algorithm, and the posterior distribution of each latent variable, as well as the mixture matrix, process noise variance, mean and covariance of stationary and non-stationary sources, are updated to obtain the trained model. The stationary source statistic of the training samples and the Wasserstein distance statistic, determined by the empirical mean and empirical covariance of the sliding window of the process uncertainty estimate, are calculated based on the training model, and the corresponding control limits are determined. Input the online running samples into the training model, calculate the online stationary source statistic and the online Wasserstein distance statistic. If either statistic exceeds the corresponding control limit, a fault is determined and an alarm message is output.

2. The robust fault detection method for the outlier-oriented non-stationary industrial process according to claim 1, characterized in that, The multivariate historical operating data includes process variables collected by at least two types of sensors, including temperature sensors, pressure sensors, flow sensors, material quantity detection devices, gas parameter detection devices, and equipment operating status detection devices. When standardizing the multivariate historical operating data, the mean and standard deviation of each process variable under normal operating conditions are used to standardize the historical samples and the online operating samples using the same parameters. The time period indicator variable satisfies: ； in, Indicates the first Does the observed sample belong to the first...? Each period, Indicates belonging to the first Each period, Indicates that it does not belong to the first Each period, This indicates the number of consecutive, non-overlapping time periods.

3. The robust fault detection method for the outlier-oriented non-stationary industrial process according to claim 1, characterized in that, The robust probabilistic stationary subspace analysis model satisfies: ； in, In the formula, For the first Standardized observation samples at each sampling time, It is an invertible mixture matrix. Let be a submatrix of a Spanning stationary subspace. Let be a submatrix of a nonstationary subspace. For potential source vectors, For a stable source, It is a non-stationary source. For process noise, The vector of outliers; The process noise satisfies: ； The dimension of the stationary source is , the dimension of the non-stationary source is , and satisfies: ； In the formula, Indicates process noise. Indicates a Gaussian distribution. Represents the process noise variance. express 3D identity matrix The dimension of the process variable representing the observed sample. Denotes the dimension of the stationary source. express 3D real space, express dimensional real space; the dimension of the stationary source This was determined using the Johansen cointegration test.

4. The robust fault detection method for the outlier-oriented non-stationary industrial process according to claim 1, characterized in that, In the first period, the stationary source mean and covariance are shared across periods, and the non-stationary source mean and covariance vary with period, specifically: ； In the formula, Indicates the first The mean of potential sources over a given period, Indicates the first Potential source covariance over time period This represents the mean of a stationary source. Indicates the covariance of stationary sources. Indicates the first The mean of non-stationary sources over a given period Indicates the first The non-stationary source covariance for each time period, with zero elements in the matrix indicating that there is no correlation between the stationary and non-stationary source groups.

5. The robust fault detection method for the outlier-oriented non-stationary industrial process according to claim 1, characterized in that, When the student t-distribution is used to model the heavy-tailed potential source, the entire time series is divided into n consecutive non-overlapping time periods. The probability density in the i-th time period is as follows: ； The probability density is further expressed as a mixture of Gaussian and gamma distributions, as shown below: ； In the formula, Indicates the first The probability density of potential sources within a given time period. Indicates the first The observation sample at the ... Potential sources corresponding to each time period Describe the t-distribution of students. Indicates the first The mean of potential sources over a given period, Indicates the first Potential source covariance over time period Indicates degrees of freedom. Represents a scale variable. Indicates a Gaussian distribution. Indicates the gamma distribution; When an observed sample deviates from the potential source distribution of its time period, the scaling variable is used to reduce the impact of that observed sample on model parameter updates.

6. The robust fault detection method for the outlier-oriented non-stationary industrial process according to claim 1, characterized in that, The variational Bayesian expectation-maximization algorithm divides the variables to be updated into a set of latent variables and a set of deterministic parameters; the set of latent variables is: ； The deterministic parameter set is as follows: ； In the formula, Represents a group of latent variables. Indicates a time period indicator variable. Indicates a potential source. Represents a scale variable. Indicates degrees of freedom. Indicates mixed weights; Represents a deterministic set of parameters. This represents the mean of a stationary source. This represents the mean of a non-stationary source. Indicates the covariance of stationary sources. This represents the covariance of a non-stationary source. Represents a mixture matrix. The variance of process noise is represented; the variational Bayesian expectation-maximization algorithm alternately executes the variational Bayesian expectation step and the variational Bayesian maximization step, wherein the variational Bayesian expectation step is used to update the posterior distribution of the latent variable set, and the variational Bayesian maximization step is used to update the deterministic parameter set.

7. A robust fault detection method for outlier nonstationary industrial processes according to claim 6, characterized in that, In the variational Bayesian maximization step, the posterior distribution of the latent sources is given by: ； in, ； ； In the formula, Let represent the optimal variational posterior distribution of the potential source. This represents the posterior mean of the potential source. This represents the potential source posterior covariance. Indicates a Gaussian distribution. Represents a mixture matrix. Represents the process noise variance. Indicates a time period indicator variable. Represents a scale variable. Indicates the first Potential source covariance over time period Indicates the first The mean of potential sources over a given period, Indicates the first One observation sample, Indicates the number of time periods. This represents the expectation under the corresponding variational distribution; In the variational Bayesian expectation step, the posterior distribution of the scale variable is given by: ； in, ； ； In the formula, This represents the optimal variational posterior distribution of the scaling variable. Represents the gamma distribution. The shape parameter representing the posterior distribution of the scaling variable. The rate parameter representing the posterior distribution of the scaling variable. Indicates degrees of freedom. Indicates the dimension of the process variable. Indicates the number of time periods. Indicates a time period indicator variable. Indicates a potential source.

8. The robust fault detection method for the outlier-oriented non-stationary industrial process according to claim 1, characterized in that, The stationary source statistic is: ； in, ； In the formula, Indicates the first The stationary source statistic for each sample. Indicates the first Potential source estimates for each sample Indicates the first Stationary source estimates for each sample. This represents the stationary source selection matrix. This represents the mean of the latent source estimates during the training phase. This represents the covariance of the potential source estimates during the training phase; The uncertainty estimate for the process is: ； The process uncertainty estimate value is collected using a sliding window of length of 1000. ； And calculate the Wasserstein distance statistic according to the following formula: ； In the formula, Indicates the first Estimated process uncertainty for a single sample Indicates the first Observations of a sample Represents a mixture matrix. Indicates the first Potential source estimates for each sample This represents the set of process uncertainty estimates within the sliding window. Indicates the length of the sliding window. Indicates the first Wasserstein distance statistic for each sample This represents the empirical mean of the process uncertainty estimates within the sliding window. The empirical covariance represents the estimate of process uncertainty within the sliding window. Indicates the dimension of the process variable. Indicates the standard deviation of process noise. Represents the trace of a matrix. This represents the L2 norm.

9. A robust fault detection method for outlier nonstationary industrial processes according to claim 8, characterized in that, Using the stationary source statistic and Wasserstein distance statistic calculated from the training samples, kernel density estimation is performed at a pre-set confidence level. Determine the control limits for stationary source statistics Control limits for Wasserstein distance statistic ; During online monitoring, the mixture matrix, process noise variance, mean of stationary sources, covariance of stationary sources, mean of non-stationary sources, and covariance of non-stationary sources in the fixed training model are inferred only for the time period indicator variables, potential sources, and scale variables corresponding to the online running samples. Fault determination meets the following criteria: ； In the formula, Indicates the preset credit level. This indicates the control limits for stationary source statistics. This indicates the control limits for the Wasserstein distance statistic. This represents the online stationary source statistics. This represents the online Wasserstein distance statistic.

10. A robust fault detection system for outlier nonstationary industrial processes, characterized in that, This includes industrial process sensors, data processing servers, and alarm terminals; The industrial process sensor is used to collect multivariable historical operating data and online operating samples of the industrial process control system under normal operating conditions. The data processing server includes a processor and a memory, wherein the memory stores a program executable by the processor, and when the program is executed by the processor, it implements the robust fault detection method as described in any one of claims 1-9. The alarm terminal is used to receive fault alarm information output by the data processing server and display the normal or fault status of the industrial process.

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