A dynamic process fault detection method based on variational Bayesian probabilistic latent variable model

Abstract

The present invention discloses a dynamic process fault detection method based on a variational Bayesian probability latent variable model, comprising: (1) constructing a probability latent variable dynamic model: collecting and standardizing industrial process data, establishing linear dynamic system representation data, and determining the initialization state prior and parameter prior of the model; (2) variational maximum step: calculating the optimal posterior distribution under the model parameters through the constructed model, and then determining the updating method of the model parameters based on the optimal posterior distribution; (3) variational expectation step: calculating the state x in the forward step n The marginal posterior distribution at each time step n is then based on the state x n Calculate the marginal posterior distribution at time step n and all previous time steps; (4) Evidence lower bound: Calculate the evidence lower bound #imgabs0# to determine that the model reaches convergence after iterative calculation; (5) Dynamic process monitoring: Construct statistics and confidence limits through the residuals of states and observations to monitor whether a fault occurs. Using the present invention, the reliability of dynamic process fault detection can be improved.

Description

Technical Field

[0001] The present invention belongs to the field of process monitoring modeling and application, and in particular relates to a dynamic process fault detection method based on a variational Bayesian probabilistic latent variable model. Background Art

[0002] Industrial processes are becoming increasingly complex and large-scale. In order to ensure the safe and stable operation of the process, it is necessary to establish a reliable monitoring model for the process. The process data shows dynamic correlation at the sampling time. The dynamic relationship of the process data makes the data no longer meet the assumption of independent and identical distribution between samples. In addition, due to the closed-loop design of industrial processes, a large number of feedback controllers are used to compensate for the loop of the system. However, the loop feedback compensation control effect of the system is difficult to predict, and the process variables are no longer executed in a deterministic manner. Industrial processes are inevitably affected by the operating environment and equipment aging, and process variables are easily affected by random noise. Therefore, traditional deterministic modeling methods face the challenge of reduced reliability.

[0003] Although linear projection dimensionality reduction methods represented by dynamic principal component analysis (DPCA) have been proposed for monitoring changes in dynamic processes, such as the Chinese patent document with publication number CN109634240A, which discloses a dynamic process monitoring method based on a new type of dynamic principal component analysis, when mining data features, it takes into account both the variance characteristics of the data and the autocorrelation characteristics of the time series, and can mine more comprehensive features.

[0004] However, this method is a deterministic model that lacks the characterization of data distribution and is easily disturbed by outliers and missing values, which reduces the reliability of the monitoring results.

[0005] In probabilistic modeling, there are also probabilistic methods represented by state space models (SSM) for modeling and monitoring dynamic processes. For example, the Chinese patent document with publication number CN113962038A discloses a method for monitoring faults in a complex mechanical system with functional cross-linking based on a heterogeneous attribute network, including: analyzing a complex mechanical system, constructing a heterogeneous attribute network model of a functional cross-linking system, and determining the dynamic attributes and static attributes in the heterogeneous attribute network model; constructing a generalized linear model GLM to connect the static and dynamic attribute distribution parameters of the heterogeneous network; determining the state space model SSM parameters using historical data; analyzing the degree of functional impact based on monitoring results; and locating the fault location.

[0006] The SSM method obtains the optimal parameters and variables through the expectation maximization (EM) method. However, as a point estimation method, the EM method only estimates the maximum a posteriori of the most likely value of each parameter. Summary of the invention

[0007] In view of the shortcomings of dynamic process fault detection methods, the present invention provides a dynamic process fault detection method based on a variational Bayesian probabilistic latent variable model, which can improve the reliability of dynamic process monitoring.

[0008] A dynamic process fault detection method based on a variational Bayesian probabilistic latent variable model comprises the following steps:

[0009] (1) Construct a probabilistic latent variable dynamic model under the variational Bayesian framework: collect industrial process data and standardize them, establish a linear dynamic system to represent the data, and determine the initial state prior and parameter prior of the model;

[0010] (2) Variational maximum step: Using the model constructed in step (1), calculate the optimal posterior distribution under the model parameters, and then determine the update method of the model parameters based on this;

[0011] (3) Variational expectation step: Based on the model constructed in step (1), the state x is calculated in the forward step n The marginal posterior distribution at each time step n is then based on the state x n Compute the marginal posterior distribution at time step n and all previous time steps;

[0012] (4) Evidence lower bound: Calculate the evidence lower bound To ensure that the model constructed by step (1) reaches convergence after iterative calculations by steps (2) and (3);

[0013] (5) Dynamic process monitoring: After the model constructed in step (1) converges, statistics and confidence limits are constructed through the residuals of states and observations to monitor whether a fault occurs.

[0014] The actual industrial process is disturbed by various noises, resulting in the randomness of the modeling system and the uncertainty of the model parameters, and the traditional dynamic process monitoring model may become inapplicable. The present invention develops a probabilistic latent variable model under the variational Bayesian framework, which represents the changes of the dynamic process through the state space equation, wherein the parameters and state variables are estimated by variational Bayesian estimation, allowing the uncertainty of the parameters to be described by probability. Specifically, the uncertain information is estimated by RTS smoothing, and the conjugate prior enables the posterior distribution of the parameters to be estimated iteratively. Therefore, the statistical characteristics of the process are represented by the mean and variance. Even in the presence of noise interference, the next state can be better predicted, and the imperceptible correlations between multiple variables can be found, so it can be well adapted to the changing system. Through the iterative update of VBEM, the parameters and state variables obtain the optimal estimate, and the uncertainty is quantified. Finally, the parameters and state variables obtained using the training data are used to construct the confidence limit of fault detection, and the online test data input model is used to detect whether the fault occurs.

[0015] In step (1), industrial process data is collected and standardized, and a linear dynamic system is established to represent the data:

[0016] Collect observation data under normal operating conditions of industrial processes n ∈R V×1 ,n=1,2,...,N as observation samples, where V is the number of variables, and the data is standardized;

[0017] Construct a linear dynamic system to represent the dynamic changes of data. Suppose the observed sample y n Through the latent state variable x n ∈R H×1 Generate, where H is the number of variables, the linear dynamic system is expressed as:

[0018]

[0019] Among them, A(H×H) is the state dynamic matrix, C(C×H) is the observation matrix, is the random variable of state evolution, is a random variable of observation noise.

[0020] Determine the model's initialization state prior and parameter prior specifically as follows:

[0021] Initial state variable x n is a Gaussian distribution with a mean of m0(H×1) and a variance of P0(H×H). The Gaussian distribution is expressed as:

[0022]

[0023] State variable x n and the observed variable y n is a Gaussian distribution, and the expected distribution of the state and observation variables is

[0024]

[0025] Among them, p(x n |x n-1 ) describes the dynamic model of the system's stochastic dynamics, p(y n |x n ) describes the probability distribution of measurement results and system states;

[0026] Automatic correlation determination ARD is performed on the hidden dimensions using Gaussian priors on the state dynamic matrix A and the observation matrix C. The system parameter prior p(θ) is constructed by the conjugate of the dynamic matrix and the noise covariance; the output noise covariance matrices Q and R are set to diagonal matrices, and the prior is defined as Q through the precision vectors τ and ρ -1 =diag(τ), R -1 =diag(ρ);

[0027] The priori representation of the system parameters is:

[0028]

[0029] Among them, a h is the h-th row vector of the state dynamic matrix A, α h is the hth element of the vector α, α and γ are the ARD parameters, is the Gamma probability density function, a α ,b α ,a γ ,b γ ,e0,g0,r0,s0 are hyper parameters.

[0030] In step (2), the optimal posterior distribution under the model parameters is calculated as:

[0031] The optimal distribution of parameters that maximizes the lower bound is expected to be at q θ The approximate posterior distribution obtained under the distribution of (θ) has the following form:

[0032]

[0033] In the formula, a h is the h-th row vector of the state dynamic matrix A, α h is the hth element of the vector α, is the Gamma probability density function, and α and γ are the ARD parameters.

[0034] In step (2), the update method of the model parameters is determined as follows:

[0035] In the α and Σ α The variational update formula is expressed as:

[0036]

[0037] In the γ and Σ γ The variational update formula is expressed as:

[0038]

[0039] In the parameters and The variational update formula is expressed as:

[0040]

[0041] In the parameters and The variational update formula is expressed as:

[0042]

[0043] In the parameters and The variational update formula is expressed as:

[0044]

[0045] In the parameters and The variational update formula is expressed as

[0046]

[0047] The specific process of step (3) is as follows:

[0048] (3-1) The variational Bayesian expectation step uses the RTS smoother to find the expected statistics of the hidden state. The RTS smoother is divided into a forward step (filtering) and a backward step (smoothing). The purpose of the forward step is to calculate the state x n At each time step n, the marginal posterior distribution, the conditional distribution is expressed as:

[0049]

[0050] (3-2) Prediction distribution q x (x n |y 1:n-1 ) is expressed as:

[0051]

[0052] (3-3) According to Schur's complement theorem, x n-1 and x n The joint distribution of the variance of

[0053]

[0054] (3-4) Quadratic form of the second-order moment T Q -1 A> is decomposed into the quadratic form and covariance of the first-order moment, expressed as

[0055] T Q -1 A>=Σ AQA + T > -1 > (15)

[0056] (3-5) Using the relationship between the precision matrix and the covariance, the covariance matrix P n-1 Expressed as

[0057]

[0058] Among them, G n-1 =(IV n-1 (I+Σ AQA V n-1 ) -1 Σ AQA ),F=G n-1 V n-1 ;

[0059] (3-6) Mean matrix m n-1 Expressed as

[0060]

[0061] (3-7) The filtering probability is expressed as

[0062]

[0063] (3-8) According to Schur's complement theorem, x n and n The joint distribution of the variance of

[0064]

[0065] (3-9) Covariance matrix S n Expressed as

[0066]

[0067] Among them, J n-1 =(IP n-1 (I+Σ CRC P n-1 ) -1 Σ CRC ),F′=J n-1 P n-1 ;

[0068] (3-10) Mean matrix Expressed as

[0069]

[0070] (3-11) Finally, the covariance matrix V is obtained by the inverse matrix of the precision matrix n

[0071]

[0072] in, is the Kalman gain;

[0073] (3-12) Mean matrix μ n Expressed as

[0074]

[0075] (3-13) In the Bayesian smoother, the marginal posterior probability at each time step is expressed as:

[0076]

[0077] (3-14)x n-1 and x n The joint distribution of the variance of

[0078]

[0079] (3-15) Variance matrix Expressed as

[0080]

[0081] in

[0082] (3-16) The mean matrix is ​​expressed as

[0083]

[0084] In the formula, μ n and V n is the forward step q x (x n |y 1:n ), m n is the predicted distribution q x (x n |y 1:n )’s mean.

[0085] In step (4), the lower bound of evidence is expressed as

[0086]

[0087] In the formula, is defined as

[0088]

[0089] Logarithmic normalization parameter Expressed as

[0090]

[0091] Among them, ξ n ,Ξ n It is expressed as:

[0092]

[0093] Among them, W n-1 =(IJ n-1 )G n-1 , μ n and V n are the mean and variance of the forward step.

[0094] The specific process of step (5) is as follows:

[0095] (5-1) The dynamic process monitoring model is divided into two parts: offline modeling and online monitoring. In the offline modeling stage, the process data under normal conditions is used to iteratively update the model parameters and state variables through the variational Bayesian model, and the confidence limit is constructed using kernel density estimation.

[0096]

[0097] Among them, K h is a non-negative kernel function; h is a smoothing parameter, called bandwidth;

[0098] (5-2) In the online monitoring phase, the residual is used to construct the fault detection statistics, T 2 The statistic is expressed as

[0099]

[0100] in,

[0101] (5-3) In the online monitoring stage, the Q statistic is expressed as

[0102]

[0103] in, r Qn is the prediction error of the Kalman filter, is the variance of the backward step.

[0104] Compared with the prior art, the present invention has the following beneficial effects:

[0105] 1. The variational Bayesian probabilistic latent variable model proposed in the present invention is used as an uncertain modeling method, in which parameters and state variables are estimated by variational Bayesian estimation.

[0106] 2. The variational Bayesian probabilistic latent variable model proposed in the present invention uses state variables to describe the dynamics of the process, making the modeling dimension much smaller than that of traditional methods.

[0107] 3. Compared with the expectation maximization method that ignores parameter uncertainty, the proposed variational Bayesian probabilistic latent variable model uses variational inference to estimate parameters, which can appropriately improve robustness.

[0108] 4. The proposed variational Bayesian probabilistic latent variable model establishes an RTS smoothing scheme under the variational Bayesian framework to estimate state variables, which can obtain the optimal state estimation under noise interference. BRIEF DESCRIPTION OF THE DRAWINGS

[0109] ​​​​Figure 1 This is a flow chart of a dynamic process fault detection method based on a variational Bayesian probabilistic latent variable model of the present invention;

[0110] Figure 2 It is a schematic diagram of the hot rolling industrial process in an embodiment of the present invention;

[0111] Figure 3 A graph showing the prediction results of the Gaussian linear dynamic system model established by the expectation maximization method in an embodiment of the present invention;

[0112] Figure 4 Graph showing prediction results of a variational Bayesian probability latent variable model in an embodiment of the present invention;

[0113] Figure 5 A process monitoring result diagram of a Gaussian linear dynamic system model established by the expectation maximization method in an embodiment of the present invention;

[0114] Figure 6 This is a diagram of process monitoring results of a variational Bayesian probabilistic latent variable model in an embodiment of the present invention;

[0115] Figure 7 A box plot of the absolute prediction error of the Gaussian linear dynamic system model established by the expectation maximization method in an embodiment of the present invention;

[0116] Figure 8 1 is a box plot of the absolute prediction error of the variational Bayesian probabilistic latent variable model in an embodiment of the present invention. DETAILED DESCRIPTION

[0117] The present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be pointed out that the embodiments described below are intended to facilitate the understanding of the present invention and do not have any limiting effect on the present invention.

[0118] like Figure 1 As shown, a dynamic process fault detection method based on a variational Bayesian probability latent variable model includes the following steps:

[0119] Step 1: Construct a linear dynamic model

[0120] (1) Collect observation data under normal operating conditions of industrial processes n ∈R V×1 ,n=1,2,...,N as observation samples, where V is the number of variables;

[0121] (2) Data standardization;

[0122] (3) Construct a linear dynamic system to represent the dynamic changes of data. Assume that the observed sample y n Through the latent state variable x n ∈R H×1Generate, where H is the number of variables, the linear dynamic system is expressed as:

[0123]

[0124] Among them, A(H×H) is the state dynamic matrix, C(C×H) is the observation matrix, is the random variable of state evolution, is a random variable of observation noise.

[0125] (4) Initial state variable x n is a Gaussian distribution with a mean of m0(H×1) and a variance of P0(H×H). The Gaussian distribution is expressed as:

[0126]

[0127] (5) State variable x n and the observed variable y n is a Gaussian distribution, and the expected distribution of the state and observation variables is

[0128]

[0129] Where p(x n |x n-1 ) describes the dynamic model of the system's stochastic dynamics, p(y n |x n ) describes the probability distribution of measurement results and system states.

[0130] (6) Automatic correlation determination (ARD) is performed on the hidden dimensions using Gaussian priors on the state dynamic matrix A and the observation matrix C. The system parameter prior p(θ) is constructed by the conjugate of the dynamic matrix and the noise covariance. The output noise covariance matrices Q and R are set to diagonal matrices, and the prior is defined as Q through the precision vectors τ and ρ -1 =diag(τ), R -1 =diag(ρ).

[0131] (7) The priori expression of system parameters is:

[0132]

[0133] Among them, a h is the h-th row vector of the state dynamic matrix A, α h is the hth element of the vector α, α and γ are the ARD parameters, is the Gamma probability density function, a α ,b α ,a γ ,b γ ,e0,g0,r0,s0 are hyper parameters.

[0134] Step 2: Variational Bayes Maximum Step

[0135] (1) The optimal distribution of the parameters that maximize the lower bound is expected to be at q θ The approximate posterior distribution has the following form:

[0136]

[0137] (2) Parameter Σ α and Σ γ The variational update formula is expressed as:

[0138]

[0139] (3) Parameters and The variational update formula is expressed as

[0140]

[0141] (4) Parameters and The variational update formula is expressed as

[0142]

[0143] Step 3: Variational Bayesian Expectation Step

[0144] (1) The variational Bayesian expectation step uses the RTS smoother to find the expected statistics of the hidden state. The RTS smoother is divided into a forward step (filtering) and a backward step (smoothing). The purpose of the forward step is to calculate the state x n At each time step n, the marginal posterior distribution, the conditional distribution is expressed as:

[0145]

[0146] (2) Prediction distribution q x (x n |y 1:n-1 ) is expressed as:

[0147]

[0148] (3) According to Schur's complement theorem, x n-1 and x n The joint distribution of the variance of

[0149]

[0150] (4) Quadratic form of the second-order moment T Q -1 ​A> is decomposed into the quadratic form and covariance of the first-order moment, expressed as

[0151] T Q -1 A>=Σ AQA + T > -1 > (46)

[0152] (5) Using the relationship between the precision matrix and the covariance, the covariance matrix P n-1 It can be expressed as

[0153]

[0154] Among them G n-1 =(IV n-1 (I+Σ AQA V n-1 ) -1 Σ AQA ),F=G n-1 V n-1 ;

[0155] (6) Mean matrix m n-1 It can be expressed as

[0156]

[0157] (7) The filtering probability is expressed as

[0158]

[0159] (8) According to Schur's complement theorem, x n and n The joint distribution of the variance of

[0160]

[0161] (9) Covariance matrix S n It can be expressed as

[0162]

[0163] Among them, J n-1 =(IP n-1 (I+Σ CRC P n-1 ) -1 Σ CRC ),F′=J n-1 P n-1 ;

[0164] (10) Mean matrix Expressed as

[0165]

[0166] (11) Finally, the covariance matrix V can be obtained by the inverse matrix of the precision matrix n

[0167]

[0168] in, is the Kalman gain.

[0169] (12) Mean matrix μ n Expressed as

[0170]

[0171] (13) In the Bayesian smoother, the marginal posterior probability at each time step is expressed as:

[0172]

[0173] (14)x n-1 and x n The joint distribution of the variance of

[0174]

[0175] (15) Variance matrix It can be expressed as

[0176]

[0177] in,

[0178] (16) The mean matrix is ​​expressed as

[0179]

[0180] Step 4: Evidence lower bound

[0181] (1) The lower bound of the evidence is expressed as

[0182]

[0183] (2) can be defined as

[0184]

[0185] (3) Logarithmic normalization parameter Expressed as

[0186]

[0187] Among them, W n-1 ,ξ n ,Ξ n It is expressed as:

[0188]

[0189] Step 5: Dynamic process monitoring

[0190] (1) The dynamic process monitoring model is divided into two parts: offline modeling and online monitoring. In the offline modeling stage, the process data under normal working conditions is used to iteratively update the model parameters and state variables through the variational Bayesian model, and the confidence limits are constructed using kernel density estimation.

[0191]

[0192] Among them, K h is a non-negative kernel function; h is a smoothing parameter, called bandwidth;

[0193] (2) In the online monitoring phase, the residual is used to construct the fault detection statistics, T 2 The statistic is expressed as

[0194]

[0195] in,

[0196] (3) In the online monitoring stage, the Q statistic is expressed as

[0197]

[0198] in,

[0199] The following is a hot rolling industrial process to illustrate the performance of the variational Bayesian probabilistic latent variable model. A typical hot strip rolling production line mainly consists of the following parts: heating furnace, roughing mill, intermediate delay roller, flying shear, finishing mill, hot output roller, laminar cooling and coiling machine. Figure 2 A schematic diagram of the described hot strip rolling industrial process is given. Due to many process rolling loops, the hot strip rolling mill involves physical quantities such as rolling force, thickness, temperature and speed. There are coupling effects of variables in the rolling process, and the change of any variable may cause a chain reaction in related operating units. In addition, the disturbance of environment and temperature makes the collected data inevitably noisy, which seriously reduces the reliability of the monitoring model.

[0200] The 1700mm hot strip rolling line was selected as the object. The process variables are the roll gap, rolling force, bending roll force and exit thickness of the 7 stands, a total of 21 variables. 1000 measured data under normal conditions were selected as training data, and 1500 measured data containing faults were selected as real-time data.

[0201] The state prediction results output by the variational Bayesian probability latent variable model are as follows: Figure 4 As shown. The Gaussian linear dynamic system model established by the expectation maximization method is shown in Figure 3 The prediction results are shown in the figure, and it can be seen that the variational Bayesian probability latent variable model proposed in the present invention has a better prediction effect.

[0202] The process monitoring results of the Gaussian linear dynamic system model established by the expectation maximization method are as follows Figure 5 As shown in the figure, the process monitoring results output by the variational Bayesian probability latent variable model are as follows: Figure 6 It can be seen that the variational Bell probability latent variable model proposed in the present invention has a more intuitive fault detection performance.

[0203] The box plot of the absolute prediction error of the Gaussian linear dynamic system model established by the expectation maximization method is shown in Figure 7 As shown, the box plot of the absolute prediction error of the variational Bayesian probability latent variable model is as follows Figure 8 It can be seen that the method proposed by the present invention has a lower prediction error.

[0204] The embodiments described above provide a detailed description of the technical solutions and beneficial effects of the present invention. It should be understood that the above are only specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, supplements and equivalent substitutions made within the scope of the principles of the present invention should be included in the protection scope of the present invention.​​​

Claims

1. A dynamic process fault detection method based on variational Bayesian probabilistic latent variable model, characterized in that: The steps include: (1) Constructing a probabilistic latent variable dynamic model under the variational Bayesian framework: Collecting industrial process data and standardizing it, establishing a linear dynamic system representation data, and determining the initialization state prior and parameter prior of the model; Specifically, collecting industrial process data and standardizing it, and establishing a linear dynamic system representation data are as follows: Collect observation data under normal operating conditions of industrial processes n ∈R V×1 , n = 1, 2, ..., N as the observation sample, where V is the number of variables, and the data is standardized; Construct a linear dynamic system to represent the dynamic changes of data. Suppose the observed sample y n Through the latent state variable x n ∈R H×1 Generate, where H is the number of variables, the linear dynamic system is expressed as: Among them, A(H×H) is the state dynamic matrix, C(C×H) is the observation matrix, is the random variable of state evolution, is a random variable of observation noise; Determine the model's initialization state prior and parameter prior specifically as follows: Initial state variable x n is a Gaussian distribution with a mean of m0(H×1) and a variance of P0(H×H). The Gaussian distribution is expressed as: State variable x n and the observed variable y n is a Gaussian distribution, and the expected distribution of the state and observation variables is Among them, p(x n |x n-1 ) describes the dynamic model of the system's stochastic dynamics, p(y n |x n ) describes the probability distribution of measurement results and system states; Automatic correlation determination ARD is performed on the hidden dimensions using Gaussian priors on the state dynamic matrix A and the observation matrix C. The system parameter prior p(θ) is constructed by the conjugate of the dynamic matrix and the noise covariance; the output noise covariance matrices Q and R are set to diagonal matrices, and the prior is defined as Q through the precision vectors τ and ρ -1 =diag(τ), R -1 =diag(ρ); The a priori representation of the system parameters is: Among them, a h is the h-th row vector of the state dynamic matrix A, α h is the hth element of the vector α, α and γ are the ARD parameters, is the Gamma probability density function, a α , b α , a γ , b γ , e0, g0, r0, s0 are hyper parameters; (2) Variational maximum step: Using the model constructed in step (1), the optimal posterior distribution under the model parameters is calculated, and then the updating method of the model parameters is determined based on the optimal posterior distribution under the model parameters. Specifically, the optimal posterior distribution under the model parameters is calculated as follows: The optimal distribution of parameters that maximizes the lower bound is expected to be at q θ The approximate posterior distribution obtained under the distribution of (θ) has the following form: In the formula, a h is the h-th row vector of the state dynamic matrix A, α h is the hth element of the vector α, is the Gamma probability density function, α and γ are the ARD parameters; The specific method for updating the model parameters is as follows: In the α and∑ α The variational update formula is expressed as: In the γ and∑ γ The variational update formula is expressed as: In the parameters and The variational update formula is expressed as: In the parameters and The variational update formula is expressed as: In the parameters and The variational update formula is expressed as: In the parameters and The variational update formula is expressed as (3) Variational expectation step: Based on the model constructed in step (1), the state x is calculated in the forward step n The marginal posterior distribution at each time step n is then based on the state x n Calculate the marginal posterior distribution at time step n and all previous time steps; the specific process is: (3-1) The variational Bayesian expectation step uses the RTS smoother to find the expected statistics of the hidden state. The RTS smoother is divided into a forward step and a backward step. The purpose of the forward step is to calculate the state x n At each time step n, the marginal posterior distribution, the conditional distribution is expressed as: (3-2) Prediction distribution q x (x n |y 1：n-1 ) is expressed as: (3-3) According to Schur's complement theorem, x n-1 and x n The joint distribution of the variance of (3-4) Quadratic form of the second-order moment Decomposed into the quadratic form and covariance of the first-order moment, expressed as (3-5) Using the relationship between the precision matrix and the covariance, the covariance matrix P n-1 Expressed as Among them, G n-1 = (IV n-1 (I+∑ AQA V n-1 ) -1 ∑ AQA ), F=G n-1 V n-1 ; (3-6) Mean matrix m n-1 Expressed as (3-7) The filtering probability is expressed as (3-8) According to Schur's complement theorem, x n and n The joint distribution of the variance of (3-9) Covariance matrix S n Expressed as Among them, J n-1 =(IP n-1 (I+∑ CRC P n-1 ) -1 ∑ CRC ), F′=J n-1 P n-1 ; (3-10) Mean matrix Expressed as (3-11) Finally, the covariance matrix V is obtained by the inverse matrix of the precision matrix n in, is the Kalman gain; (3-12) Mean matrix μ n Expressed as (3-13) In the Bayesian smoother, the marginal posterior probability at each time step is expressed as: (3-14)x n-1 and x n The joint distribution of the variance of (3-15) Variance matrix Expressed as in (3-16) The mean matrix is ​​expressed as In the formula, μ n and V n is the forward step q x (x n |y 1：n ), m n is the predicted distribution q x (x n |y 1：n )’s mean; (4) Evidence lower bound: Calculate the evidence lower bound To determine whether the model constructed by step (1) reaches convergence after iterative calculations of steps (2) and (3); the lower bound of the evidence is expressed as In the formula, is defined as Logarithmic normalization parameter Expressed as Among them, ξ n ,Ξ n It is expressed as: Among them, W n-1 =(IJ n-1 )G n-1 , μ n and V n are the mean and variance of the forward step; (5) Dynamic process monitoring: After the model constructed in step (1) converges, statistics and confidence limits are constructed through the residuals of the state and observation to monitor whether a fault occurs. The specific process is as follows: (5-1) The dynamic process monitoring model is divided into two parts: offline modeling and online monitoring. In the offline modeling stage, the process data under normal conditions is used to iteratively update the model parameters and state variables through the variational Bayesian model, and the confidence limit is constructed using kernel density estimation. Among them, K h is a non-negative kernel function; h is a smoothing parameter, called bandwidth; (5-2) In the online monitoring phase, the residual is used to construct the fault detection statistics, T 2 The statistic is expressed as in, (5-3) In the online monitoring stage, the Q statistic is expressed as in, is the prediction error of the Kalman filter, is the variance of the backward step.

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