Intermittent process monitoring method based on probability slow feature analysis and partial least squares

Abstract

The invention discloses an intermittent process monitoring method based on probability slow feature analysis and partial least squares, and belongs to the technical field of intermittent process monitoring. The method comprises the following steps: an off-line modeling stage: preprocessing normal working condition sample data, establishing a statistical model, and determining a control limit of SPE statistical magnitude; in the online monitoring stage, data collected in real time are preprocessed in the same way as offline modeling, and then the data are input into the trained statistical model to calculate the sum SPE statistical magnitude; and if the statistical magnitude exceeds the set control limit, judging that the system breaks down. Compared with a traditional multivariate statistical method, a process monitoring model based on probability slow feature analysis and partial least squares can realize quality-related process monitoring and more timely and accurately monitor the occurrence of faults, so that an alarm is given in time, and the production efficiency is improved; and establishing a cobalt oxalate synthesis process simulation platform by utilizing MATLAB and generating data, and carrying out simulation verification on the effectiveness of the method.

Description

Technical Field

[0001] This invention relates to the field of intermittent process monitoring technology, specifically to an intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares. Background Technology

[0002] Batch production processes, as an indispensable method of industrial production, are widely used in fields closely related to daily life, such as pharmaceuticals, food processing, metal processing, and fine chemicals. In recent years, with the rapid development of modern society, the market demand for diversified, personalized, and high-quality products has been continuously rising, prompting modern industrial production to increasingly rely on batch production processes capable of producing small batches of high-value-added products. Under these circumstances, ensuring the safe and stable operation of batch production processes and pursuing high-quality products have become core concerns in industrial production.

[0003] Existing process monitoring methods still suffer from the following problems: Traditional multivariate statistical process monitoring methods often trigger alarms by monitoring deviations from normal operating conditions. However, industrial data often exhibits dynamic and slowly time-varying characteristics, making it difficult for monitoring methods to accurately distinguish between operating condition changes and actual faults. Furthermore, traditional multivariate statistical methods cannot extract quality-related process features, thus failing to achieve quality-related process monitoring and consequently, failing to determine whether current abnormal operating conditions truly affect product quality.

[0004] Furthermore, due to the slow time-varying nature of intermittent processes, although slow feature analysis can use the magnitude of the first derivative of the feature with respect to time to measure the rate of change of the feature, further separating noise and feature data and extracting the slowest-changing potential features, the process trend under closed-loop control exhibits a slowly changing sequential phase.

[0005] Therefore, ensuring the safe and stable operation of intermittent production processes and pursuing high-quality products, as well as accurately distinguishing between operating condition switching and actual faults, are problems that urgently need to be solved by those skilled in the art. In response, we propose an intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares. Summary of the Invention

[0006] The purpose of this invention is to provide a method for monitoring intermittent processes based on probabilistic slow feature analysis and partial least squares. This method involves preprocessing normal operating condition sample data, establishing a statistical model, and determining... Control limits for the SPE statistic; during the online monitoring phase, the real-time collected data undergoes the same preprocessing as in offline modeling before being input into the trained statistical model to calculate... The method involves calculating the SPE statistic; if the statistic exceeds the set control limit, the system is considered to have malfunctioned. This invention utilizes a process monitoring model based on probabilistic slow feature analysis and partial least squares, which, compared to traditional multivariate statistical methods, enables quality-related process monitoring, more timely and accurate detection of faults, and timely alarm issuance, thereby improving production efficiency. Furthermore, a simulation platform for the cobalt oxalate synthesis process is established using MATLAB, and data is generated to verify the effectiveness of the proposed method, thus solving the problems mentioned in the background section.

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares includes the following steps:

[0009] S1: Obtain intermittent process variable data and perform standardization processing;

[0010] S2: Use probabilistic slow feature analysis to extract slow features from the standardized data;

[0011] S3: Use partial least squares method to establish a monitoring model for intermittent processes;

[0012] S4: Calculate the monitoring statistics and determine the control limits. Determine whether the statistics exceed the control limits. If they exceed the control limits, it indicates a fault has occurred; otherwise, it indicates normal operation.

[0013] Further, step S1 includes the following steps:

[0014] The sample for collecting intermittent process data is , where the symbol , , These represent the number of batches measured, process variables, and sampling times, respectively, with batch data constructed in a three-dimensional data format;

[0015] Expanding the data in batches according to the time series, the expanded matrix representation is as follows:

[0016] (1)

[0017] For the expanded matrix To standardize it, its expression is:

[0018] (2)

[0019] in, Indicates the extended Data matrix of batches; Indicates the first The first batch The sampling time of the first time One value; Indicates to Standardized data; express The mean; express The variance.

[0020] Further, step S2 includes the following steps:

[0021] Standardized data Decomposed into time series observations , Observation vector at time As input to the observation model, its monitoring model expression is:

[0022] (3)

[0023] in, Represents a diagonal matrix; This represents the state transition matrix, indicating that the slow features at the current time step are influenced by the slow features at the previous time step. The state emission matrix represents the observed data. Through slow features To express; Indicates state noise; Indicates measurement noise; The mean is 0 and the variance is . The normal distribution; The mean is 0 and the variance is . The normal distribution; This represents the variance of the state transition noise term; This represents the variance of the state emission noise term;

[0024] in, The expression for slowness in the equation is:

[0025] (4)

[0026] in, Indicates the first A slow feature; Represents the state transition matrix The 1 eigenvalue; when When =1, express (t)= (t-1), (t-1) represents the slow characteristic at time t-1; (t) represents the slow characteristic at time t, which is the slowest change.

[0027] Furthermore, step S2 also includes the following steps:

[0028] The monitoring model needs to determine the parameters. Based on the first-order Markov assumption and the observation independence assumption, the log-likelihood function of the complete data is expressed as:

[0029] (5)

[0030] in, This represents the set of observation data at all time steps. Represents the set of hidden state sequences for all time steps;

[0031] Within the framework of the Expectation-Maximization (EM) algorithm, the model parameters are solved by calculating the conditional expectation of the log-likelihood function of the complete data with respect to the observed data. Represented as:

[0032] (6)

[0033] In the formula, This represents the parameters of the previous iteration of the EM algorithm;

[0034] Expanding equation (6) yields:

[0035]

[0036] (7)

[0037] in, Represents a constant term that is independent of the parameter; The dimension representing slow features; This represents the posterior expectation operator.

[0038] Furthermore, step S2 also includes the following steps:

[0039] By iteratively optimizing the function, the lower bound of the evidence of the monitoring model gradually approaches the maximum value until the preset iteration termination criterion is reached;

[0040] The parameter estimation problem in the EM algorithm is solved by iterating between the expected step (E-step) and the maximum step (M-step). After the EM algorithm iteration, each parameter is obtained.

[0041] In step E, the expectation of equation (7) is calculated:

[0042] (8)

[0043] in, State vector The posterior expectation; This represents the smoothed mean of the reverse smoothed output. Indicates time and time The expectation of the cross-correlation matrix of the state vectors; Represents the smoothing gain matrix; Indicates time The smoothed state covariance; This represents the expected value of the autocorrelation matrix of the state vector at any given moment;

[0044] The forward recursive derivation yields Based on the posterior distribution of the currently known observation data, its expression is:

[0045] (9)

[0046] The forward recursive derivation formula is as follows:

[0047] (10)

[0048] in, Represent the prior covariance matrix; Represents the Kalman gain matrix; This represents the mean of the posterior states; Represent the posterior covariance matrix; express The posterior covariance matrix at time 1; the forward recursion completes to obtain the values ​​from time 1 to time 2. The forward and backward distribution at time 1;

[0049] when When the backward posterior distribution is the same as the forward posterior distribution, then...

[0050] (11)

[0051] The reverse smoothing formula is:

[0052] (12)

[0053] in, Represents the smoothing gain matrix; This represents the mean of a smoothed state. This represents the smooth state covariance; it can be calculated through forward and backward recursive iterations. The expected value of the slow feature required by the function.

[0054] Furthermore, step S2 also includes the following steps:

[0055] In the M-step, partial derivatives are calculated and set to zero to update the model parameters. ;

[0056] Solve for parameters and Its expression is:

[0057] (13)

[0058] in,

[0059] (14)

[0060] in, This represents the negative cumulative amount of state correlation. Represents the correction amount for state energy; This represents a symmetric correlation constraint; Indicates time span constraints;

[0061] For parameters It can be updated to:

[0062] (15)

[0063] For parameters It can be updated to:

[0064] (16)

[0065] in, Indicates the noise variance; The first element of the observation matrix is... OK;

[0066] After updating the monitoring model parameters as described above, the extracted slow features are obtained.

[0067] Further, step S3 includes the following steps:

[0068] Based on step S2, obtain the extracted slow features. Enter data at this time have There are samples, with dimensions of Output data Dimensions PLS reduces high-dimensional data to a low-dimensional space composed of latent variables. Before modeling, the data needs to be standardized to ensure that its mean is 0 and its variance is 1. After PLS decomposition, the input and output data can be represented as follows:

[0069] (17)

[0070] in, Indicates the first A score vector; express The load vector; express The load vector; and They represent and The load matrix; Represents the score matrix; Indicates the number of latent variables; and These are the residual matrices for the input and output, respectively; , , , They represent respectively to , , , The transpose of .

[0071] Furthermore, step S3 also includes the following steps:

[0072] The nonlinear iterative partial least squares (NIPALS) algorithm is used to solve for PLS. Its goal is to maximize the covariance of each set of latent variables under the constraints, as shown below:

[0073] (18)

[0074] in, and Represents the weight vector; score vector. , It can be calculated as:

[0075] (19).

[0076] Further, step S4 includes the following steps:

[0077] Select The SPE statistic is used to evaluate the performance of the monitoring model. The sum of squares of the score vectors is used to measure the degree to which a sample deviates from the model. obey Distribution, for the th Each sampling time, The statistic is defined as:

[0078] (20)

[0079] in, The covariance matrix representing the score at this moment; Indicates the sampling batch; Indicates the first Slow feature score vectors for each batch; The reference mean vector representing the slow feature score vector; Represents the covariance matrix The inverse matrix of is used to calculate the control limits, and its expression is:

[0080] (twenty one)

[0081] in, express The diagonal elements; and These represent the degrees of freedom in the numerator and the degrees of freedom in the denominator, respectively.

[0082] Furthermore, step S4 also includes the following steps:

[0083] SPE statistic represents The degree of deviation of the measured value from the established monitoring model at any given time represents the changes in the data that are not explained by the model.

[0084] When the SPE exceeds the control limit, a fault is considered to have occurred in the process. The SPE statistic is specifically calculated as follows:

[0085] (twenty two)

[0086] , (twenty three)

[0087] in, Indicates the first The sum of squared reconstruction errors for each sample; Indicates the first The sample at the th Residuals on each variable; Indicates the scaling factor; Indicates degrees of freedom; Describing the degrees of freedom as The chi-square distribution; Indicates the first Sample mean estimation for each sample SPE; Indicates the first Sample variance estimation for each sample SPE.

[0088] Compared with the prior art, the beneficial effects of the present invention are:

[0089] In this invention, a statistical model is established using sample data from normal operating conditions, and the determination is made. Control limits for the SPE statistic; during the online monitoring phase, the real-time collected data undergoes the same preprocessing as in offline modeling before being input into the trained statistical model to calculate... The system uses SPE statistics; if the statistics exceed the set control limits, the system is judged to have failed. Compared with traditional process monitoring models, the advantages of this invention are that it can realize quality-related process monitoring, accurately distinguish between working condition switching and actual faults; it can also perform slow feature extraction on intermittent process data for process monitoring, thereby timely detecting faults and issuing alarms, further reducing losses and improving production efficiency. Attached Figure Description

[0090] Figure 1 This is a flowchart of the intermittent process monitoring process of the present invention;

[0091] Figure 2 The step fault in the cobalt oxalate synthesis process of this invention Statistical monitoring results chart;

[0092] Figure 3 This is a graph showing the monitoring results of the step fault SPE statistics in the cobalt oxalate synthesis process of the present invention. Detailed Implementation

[0093] 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 some embodiments of the present invention, and not all embodiments. 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.

[0094] To address the challenges of ensuring safe and stable operation of intermittent production processes, pursuing high-quality products, and accurately distinguishing between operating condition changes and actual faults in existing technologies, please refer to [link to relevant documentation]. Figures 1-3 This embodiment provides the following technical solution:

[0095] The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares includes the following steps:

[0096] S1: Obtain intermittent process variable data and perform standardization processing;

[0097] S2: Use probabilistic slow feature analysis to extract slow features from the standardized data;

[0098] S3: Use partial least squares method to establish a monitoring model for intermittent processes;

[0099] S4: Calculate the monitoring statistics and determine the control limits. Determine whether the statistics exceed the control limits. If they exceed the control limits, it indicates a fault has occurred; otherwise, it indicates normal operation.

[0100] Repeat steps S1-S4 until all samples have been monitored.

[0101] Since the synthesis of cobalt oxalate is a classic batch process, it is widely used in process monitoring and quality prediction. Therefore, this invention describes the specific implementation method in conjunction with the synthesis process of cobalt oxalate.

[0102] The cobalt oxalate synthesis process was simulated using MATLAB. The key process variables selected in this invention are temperature, stirring rate, ammonium oxalate flow rate, ammonium oxalate concentration, initial volume of cobalt chloride, and cobalt chloride concentration. The selected process variables and their parameter values ​​are shown in Table 1.

[0103] Table 1. Selected Process Variables and Their Parameter Setting Ranges

[0104] Process variables value temperature( ) 321 329 <![CDATA[ Concentration (mol / m 3 )]]> 1670.1 1701.1 <![CDATA[Initial volume of CoCl2 (m 3 )]]> 1.38 1.42 <![CDATA[ Flow velocity (m) 3 / s)]]> 1.5 2.2 <![CDATA[Initial volume of CoCl2 (mol / m 3 )]]> 1.1 1.2 Stirring rate (rps) 4.00 5.00

[0105] Step 1: Collect samples of intermittent process data , where the symbol , , These represent the number of batches measured, process variables, and sampling times, respectively, with batch data constructed in a three-dimensional data format;

[0106] Expanding the data in batches according to the time series, the expanded matrix representation is as follows:

[0107] (1)

[0108] For the expanded matrix To standardize it, its expression is:

[0109] (2)

[0110] in, Indicates the extended Data matrix of batches; Indicates the first The first batch The sampling time of the first time One value; Indicates to Standardized data; express The mean; express The variance.

[0111] Step 2: Standardize the data Decomposed into time series observations , Observation vector at time As input to the observation model, probabilistic slow feature analysis is a probabilistic extension of slow feature analysis. It can consider the impact of noise when extracting slow features and describe the extraction process from a probabilistic perspective. Its monitoring model expression is as follows:

[0112] (3)

[0113] in, Represents a diagonal matrix; This represents the state transition matrix, indicating that the slow features at the current time step are influenced by the slow features at the previous time step. The state emission matrix represents the observed data. Through slow features To express; Indicates state noise; Indicates measurement noise; The mean is 0 and the variance is . The normal distribution; The mean is 0 and the variance is . The normal distribution; This represents the variance of the state transition noise term; This represents the variance of the state emission noise term;

[0114] in, The expression for slowness in the equation is:

[0115] (4)

[0116] in, Indicates the first A slow feature; Represents the state transition matrix The 1 eigenvalue; when When =1, express (t)= (t-1), (t-1) represents the slow characteristic at time t-1; (t) represents the slow characteristic at time t, which is the slowest change.

[0117] The monitoring model needs to determine the parameters. Based on the first-order Markov assumption and the observation independence assumption, the log-likelihood function of the complete data is expressed as:

[0118] (5)

[0119] in, This represents the set of observation data at all time steps. Represents the set of hidden state sequences for all time steps;

[0120] Within the framework of the Expectation-Maximization (EM) algorithm, the model parameters are solved by calculating the conditional expectation of the log-likelihood function of the complete data with respect to the observed data. Represented as:

[0121] (6)

[0122] In the formula, This represents the parameters of the previous iteration of the EM algorithm;

[0123] Expanding equation (6) yields:

[0124]

[0125] (7)

[0126] in, Represents a constant term that is independent of the parameter; The dimension representing slow features; Represents the posterior expectation operator;

[0127] By iteratively optimizing the function, the lower bound of the evidence of the monitoring model gradually approaches the maximum value until the preset iteration termination criterion is reached;

[0128] The parameter estimation problem in the EM algorithm is solved by iterating between the expected step (E-step) and the maximum step (M-step). After the EM algorithm iteration, each parameter is obtained.

[0129] In step E, the expectation of equation (7) is calculated:

[0130] (8)

[0131] in, State vector The posterior expectation; This represents the smoothed mean of the reverse smoothed output. Indicates time and time The expectation of the cross-correlation matrix of the state vectors; Represents the smoothing gain matrix; Indicates time The smoothed state covariance; This represents the expected value of the autocorrelation matrix of the state vector at any given moment;

[0132] The forward recursive derivation yields Based on the posterior distribution of the currently known observation data, its expression is:

[0133] (9)

[0134] The forward recursive derivation formula is as follows:

[0135] (10)

[0136] in, Represent the prior covariance matrix; Represents the Kalman gain matrix; This represents the mean of the posterior states; Represent the posterior covariance matrix; express The posterior covariance matrix at time 1; the forward recursion completes to obtain the values ​​from time 1 to time 2. The forward and backward distribution at time t.

[0137] when When the backward posterior distribution is the same as the forward posterior distribution, then...

[0138] (11)

[0139] The reverse smoothing formula is:

[0140] (12)

[0141] in, Represents the smoothing gain matrix; This represents the mean of a smoothed state. This represents the smooth state covariance; it can be calculated through forward and backward recursive iterations. The expected value of the slow feature required by the function.

[0142] In the M-step, partial derivatives are calculated and set to zero to update the model parameters. ;

[0143] Solve for parameters and Its expression is:

[0144] (13)

[0145] in,

[0146] (14)

[0147] in, This represents the negative cumulative amount of state correlation. Represents the correction amount for state energy; This represents a symmetric correlation constraint; Indicates time span constraints;

[0148] For parameters It can be updated to:

[0149] (15)

[0150] For parameters It can be updated to:

[0151] (16)

[0152] in, Indicates the noise variance; The first element of the observation matrix is... OK;

[0153] After updating the monitoring model parameters as described above, the extracted slow features are obtained.

[0154] Step 3: Based on step S2, obtain the extracted slow features. Enter data at this time have There are samples, with dimensions of Output data Dimensions PLS reduces high-dimensional data to a low-dimensional space composed of latent variables. Before modeling, the data needs to be standardized to ensure that its mean is 0 and its variance is 1. After PLS decomposition, the input and output data can be represented as follows:

[0155] (17)

[0156] in, Indicates the first A score vector; express The load vector; express The load vector; and They represent and The load matrix; Represents the score matrix; Indicates the number of latent variables; and These are the residual matrices for the input and output, respectively; , , , They represent respectively to , , , The transpose of .

[0157] The nonlinear iterative partial least squares (NIPALS) algorithm is used to solve for PLS. Its goal is to maximize the covariance of each set of latent variables under the constraints, as shown below:

[0158] (18)

[0159] in, and Represents the weight vector; score vector. , It can be calculated as:

[0160] (19).

[0161] Step 4: Selection The SPE statistic is used to evaluate the performance of the monitoring model. The sum of squares of the score vectors is used to measure the degree to which a sample deviates from the model. obey Distribution, for the th Each sampling time, The statistic is defined as:

[0162] (20)

[0163] in, The covariance matrix representing the score at this moment; Indicates the sampling batch; Indicates the first Slow feature score vectors for each batch; The reference mean vector representing the slow feature score vector; Represents the covariance matrix The inverse matrix of is used to calculate the control limits, and its expression is:

[0164] (twenty one)

[0165] in, express The diagonal elements; and These represent the degrees of freedom in the numerator and the degrees of freedom in the denominator, respectively.

[0166] SPE statistic represents The degree of deviation of the measured value from the established monitoring model at any given time represents the changes in the data that are not explained by the model.

[0167] When the SPE exceeds the control limit, a fault is considered to have occurred in the process. The SPE statistic is specifically calculated as follows:

[0168] (twenty two)

[0169] , (twenty three)

[0170] in, Indicates the first The sum of squared reconstruction errors for each sample; Indicates the first The sample at the th Residuals on each variable; Indicates the scaling factor; Indicates degrees of freedom; Describing the degrees of freedom as The chi-square distribution; Indicates the first Sample mean estimation for each sample SPE; Indicates the first Sample variance estimation for each sample SPE.

[0171] Data on the real-time operation of the cobalt oxalate synthesis process was obtained according to step 1 and standardized. Slow features were extracted from the standardized data using probabilistic slow feature analysis according to step 2. An offline process monitoring model was established using partial least squares analysis based on the slow feature data according to step 3 for process monitoring. The process was then calculated under normal operating conditions according to step 4. And SPE control limits, and calculate monitoring data. The SPE statistic is calculated, and it is determined whether the statistic exceeds the control limit. If it exceeds the control limit, it indicates that a fault has occurred; if it does not exceed the control limit, it indicates that the condition is normal. At this time, return to step 1 to monitor the next sample.

[0172] To verify the effectiveness of the proposed method for online monitoring of batch processes, a total of 1500 batches of process data were generated using a cobalt oxalate synthesis process simulation platform. The fault type was a step fault with a 10% increase in ammonium oxalate flow rate. The first 1200 batches were data generated under normal operating conditions, and batches 1201 to 1500 were fault-prone batches. The first 600 batches of normal sample data served as the training set, and the last 600 batches of normal sample data and 300 batches of fault data served as the test set, simulating the occurrence of faults under real operating conditions. The relevant process monitoring results are as follows: Figures 2-3 As shown.

[0173] The beneficial effects achieved by the above content are as follows: Compared with traditional process monitoring models, the advantages of this invention are that it can realize quality-related process monitoring, accurately distinguish between working condition switching and actual faults; it can also perform slow feature extraction on intermittent process data for process monitoring, thereby timely detecting faults and issuing alarms, further reducing losses and improving production efficiency.

[0174] Working principle: By using sample data from normal operating conditions to establish a statistical model and determine... Control limits for the SPE statistic; during the online monitoring phase, the real-time collected data undergoes the same preprocessing as in offline modeling before being input into the trained statistical model to calculate... And SPE statistics; if the statistics exceed the set control limits, the system is judged to have malfunctioned.

[0175] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "possessing," or any other variations thereof are intended to cover non-exclusive possession, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus.

[0176] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that variations, modifications, substitutions and alterations may be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for monitoring intermittent processes based on probabilistic slow feature analysis and partial least squares, characterized in that, Includes the following steps: S1: Obtain intermittent process variable data and perform standardization processing; S2: Use probabilistic slow feature analysis to extract slow features from the standardized data; S3: Use partial least squares method to establish a monitoring model for intermittent processes; S4: Calculate the monitoring statistics and determine the control limits. Determine whether the statistics exceed the control limits. If they exceed the control limits, it indicates a fault has occurred; otherwise, it indicates normal operation.

2. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares as described in claim 1, characterized in that, Step S1 includes the following steps: The sample for collecting intermittent process data is , where the symbol , , These represent the number of batches measured, process variables, and sampling times, respectively, with batch data constructed in a three-dimensional data format; Expanding the data in batches according to the time series, the expanded matrix representation is as follows: （1） For the expanded matrix To standardize it, its expression is: （2） in, Indicates the extended Data matrix of batches; Indicates the first The first batch The sampling time of the first time One value; Indicates to Standardized data; express The mean; express The variance.

3. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares as described in claim 2, characterized in that, Step S2 includes the following steps: Standardized data Decomposed into time series observations , Observation vector at time As input to the observation model, the monitoring model expression is: （3） in, Represents a diagonal matrix; This represents the state transition matrix, indicating that the slow features at the current time step are influenced by the slow features at the previous time step. The state emission matrix represents the observed data. Through slow features To express; Indicates state noise; Indicates measurement noise; The mean is 0 and the variance is . The normal distribution; The mean is 0 and the variance is . The normal distribution; This represents the variance of the state transition noise term; This represents the variance of the state emission noise term; in, The expression for slowness in the equation is: (4) in, Indicates the first A slow feature; Represents the state transition matrix The 1 eigenvalue; when When =1, express (t)= (t-1), (t-1) represents the slow characteristic at time t-1; (t) represents the slow characteristic at time t, which is the slowest change.

4. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares according to claim 3, characterized in that, Step S2 also includes the following steps: Monitoring model determines parameters Based on the first-order Markov assumption and the observation independence assumption, the log-likelihood function of the complete data is expressed as: (5) in, This represents the set of observation data at all time steps. Represents the set of hidden state sequences for all time steps; Within the framework of the Expectation-Maximization (EM) algorithm, the model parameters are solved by calculating the conditional expectation of the log-likelihood function of the complete data with respect to the observed data. Represented as: （6） In the formula, This represents the parameters of the previous iteration of the EM algorithm; Expanding equation (6) yields: (7) in, Represents a constant term that is independent of the parameter; The dimension representing slow features; This represents the posterior expectation operator.

5. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares according to claim 4, characterized in that, Step S2 also includes the following steps: By iteratively optimizing the function, the lower bound of the evidence of the monitoring model gradually approaches the maximum value until the preset iteration termination criterion is reached; The parameter estimation problem in the EM algorithm is solved by iterating between the expected step and the maximum step. After the EM algorithm iteration, each parameter is obtained. In step E, the expectation of equation (7) is calculated: (8) in, State vector The posterior expectation; This represents the smoothed mean of the reverse smoothed output. Indicates time and time The expectation of the cross-correlation matrix of the state vectors; Represents the smoothing gain matrix; Indicates time The smoothed state covariance; This represents the expected value of the autocorrelation matrix of the state vector at any given moment; The forward recursive derivation yields Based on the posterior distribution of the currently known observation data, the expression is: (9) The forward recursive derivation formula is as follows: (10) in, Represent the prior covariance matrix; Represents the Kalman gain matrix; This represents the mean of the posterior states; Represent the posterior covariance matrix; express The posterior covariance matrix at time 1; the forward recursion completes to obtain the values ​​from time 1 to time 2. The forward and backward distribution at time 1; when When the backward posterior distribution is the same as the forward posterior distribution, then... (11) The reverse smoothing formula is: (12) in, Represents the smoothing gain matrix; This represents the mean of a smoothed state. Represents the smooth state covariance; calculated through forward and backward recursive iterations. The expected value of the slow feature required by the function.

6. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares according to claim 5, characterized in that, Step S2 also includes the following steps: In the M-step, partial derivatives are calculated and set to zero to update the model parameters. ; Solve for parameters and Its expression is: （13） in, (14) in, This represents the negative cumulative amount of state correlation. Represents the correction amount for state energy; This represents a symmetric correlation constraint; Indicates time span constraints; For parameters It can be updated to: (15) For parameters It can be updated to: （16） in, Indicates the noise variance; The first element of the observation matrix is... OK; After updating the monitoring model parameters, the extracted slow features are obtained.

7. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares according to claim 6, characterized in that, Step S3 includes the following steps: Based on step S2, obtain the extracted slow features. Enter data at this time have There are samples, with dimensions of Output data Dimensions PLS reduces high-dimensional data to a low-dimensional space composed of latent variables. Before modeling, the data needs to be standardized to ensure a mean of 0 and a variance of 1. After PLS decomposition, the input and output data can be represented as follows: （17） in, Indicates the first A score vector; express The load vector; express The load vector; and They represent and The load matrix; Represents the score matrix; Indicates the number of latent variables; and These are the residual matrices for the input and output, respectively; , , , They represent respectively to , , , The transpose of .

8. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares according to claim 7, characterized in that, Step S3 also includes the following steps: The PLS is solved using a nonlinear iterative partial least squares algorithm, with the goal of maximizing the covariance of each set of latent variables under the constraints, as shown below: （18） in, and Represents the weight vector; score vector. , It can be calculated as: （19）。 9. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares according to claim 8, characterized in that, Step S4 includes the following steps: Select The SPE statistic is used to evaluate the performance of the monitoring model. The sum of squares of the score vectors is used to measure the degree to which a sample deviates from the model. obey Distribution, for the th Each sampling time, The statistic is defined as: （20） in, The covariance matrix representing the score at this moment; Indicates the sampling batch; Indicates the first Slow feature score vectors for each batch; The reference mean vector representing the slow feature score vector; Represents the covariance matrix The inverse matrix of is used to calculate the control limits, and its expression is: （21） in, express The diagonal elements; and These represent the degrees of freedom of the numerator and the degrees of freedom of the denominator, respectively.

10. The intermittent process monitoring method based on probabilistic slow feature analysis and partial least squares according to claim 9, characterized in that, Step S4 also includes the following steps: SPE statistic represents The degree of deviation of the measured value from the established monitoring model at any given time represents the changes in the data that are not explained by the model. When the SPE exceeds the control limit, a fault is considered to have occurred in the process. The SPE statistic is specifically calculated as follows: （22） , （23） in, Indicates the first The sum of squared reconstruction errors for each sample; Indicates the first The sample at the th Residuals on each variable; Indicates the scaling factor; Indicates degrees of freedom; Describing the degrees of freedom as The chi-square distribution; Indicates the first Sample mean estimation for each sample SPE; Indicates the first Sample variance estimation for each sample SPE.

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