A distributed running state monitoring method based on local manifold structure and slow features

Abstract

The application discloses a distributed running state monitoring method based on local manifold structure and slow features, which divides input process variables into four blocks of Gaussian dynamic, Gaussian non-dynamic, non-Gaussian dynamic and non-Gaussian non-dynamic according to different variable types, adopts SFA method for the dynamic part, adopts NPE method for the non-dynamic part, considers the correlation between the blocks, introduces mutual information of the blocks, and finally obtains the final running state monitoring result through local outlier factors to comprehensively integrate the information in the blocks and between the blocks. The application not only improves the accuracy of the running state monitoring, but also enhances the adaptability to complex dynamic processes, and provides effective support for stable running and intelligent maintenance of industrial equipment.

Description

Technical fields:

[0001] This invention relates to a distributed operation status monitoring method based on local manifold structure and slow characteristics. Background technology:

[0002] In modern industry, the research and application of operational status monitoring technology are becoming increasingly important, especially in production and equipment maintenance. Data-driven operational status monitoring methods, due to their efficiency and accuracy, are gradually becoming mainstream. These methods utilize large amounts of historical and real-time monitoring data, employing machine learning and statistical analysis techniques to automatically identify and diagnose equipment faults. Compared to traditional model-based methods, data-driven methods do not rely on physical or mathematical models and can better handle uncertainties in complex systems. However, when processing high-dimensional data, data-driven methods often face challenges such as the curse of dimensionality and feature redundancy, thus requiring effective feature extraction and dimensionality reduction techniques.

[0003] Neighborhood Preserving Embedding (NPE) and Slow Feature Analysis (SFA) are two techniques that have received widespread attention in the fields of data dimensionality reduction and feature extraction in recent years. NPE primarily achieves low-dimensional feature mapping by preserving neighborhood relationships, effectively retaining local structural information of the data and making it suitable for processing nonlinear data. SFA, on the other hand, focuses on extracting slowly changing features from data, capturing long-term trends and potential patterns. Applying these two techniques to the field of operational status monitoring can effectively extract and reduce the dimensionality of data, thereby improving the accuracy of operational status monitoring and providing strong support for intelligent maintenance of industrial equipment. Summary of the Invention:

[0004] The technical problem this invention aims to solve is that traditional operational status monitoring methods often fail to effectively adapt to diverse variable characteristics when dealing with complex industrial process variables, especially when data exhibits different statistical properties, leading to insufficient accuracy and reliability in operational status monitoring. Specifically, process variables can be categorized into four types based on their statistical characteristics: Gaussian dynamic, Gaussian non-dynamic, non-Gaussian dynamic, and non-Gaussian non-dynamic. However, existing operational status monitoring technologies typically employ a uniform approach for all variables, failing to fully utilize their unique characteristics, resulting in information loss and performance degradation. Therefore, this invention proposes a novel operational status monitoring method that subdivides the dataset into these four categories. This allows for flexible selection of appropriate fault detection techniques (NPE or SFA) based on the characteristics of each variable type, and also enables the calculation of mutual information between blocks, considering the correlation between variables across blocks, thus achieving more accurate operational status monitoring. NPE can effectively preserve the local structure of high-dimensional data, while SFA can capture slowly changing features. By combining these two methods and incorporating mutual information between different sub-blocks, this invention not only improves the accuracy of operational status monitoring but also enhances its adaptability to complex dynamic processes, providing effective support for the stable operation and intelligent maintenance of industrial equipment. The specific steps are as follows:

[0005] Step (1): Data acquisition and data preprocessing, the specific implementation process is as follows:

[0006] Step (1.1): Collect monitoring data from m sensors at n sampling times under normal operating conditions to obtain the dataset X = [x1, x2, ..., x...]. n ] T ∈R n*m , where n is the number of samples in the dataset, m is the number of sensors, and R is the set of real numbers.

[0007] Step (1.2): Preprocess the data using the z-score normalization method, as shown in the following formula:

[0008]

[0009] Where x i It is a sample collected by a sensor, μ i It is the sample mean, σ i The standard deviation is the standard deviation. The dataset X is obtained after standardizing the data in each column. train ∈R n*m ;

[0010] Step (2): According to X train The subspace is divided based on whether different variables in X exhibit dynamic characteristics and whether they follow a Gaussian distribution. trainIt is divided into four sub-blocks: Gaussian non-dynamic block X1, non-Gaussian non-dynamic block X2, Gaussian dynamic block X3, and non-Gaussian dynamic block X4. The specific steps are as follows:

[0011] Step (2.1): The division of dynamic and non-dynamic subspaces is based on the temporal correlation of the variables themselves. That is, if a variable has temporal correlation, it is classified as a dynamic variable; otherwise, it is classified as a non-dynamic variable. Different types of x i ∈R n×1 (i = 1, 2, ..., m) exhibit varying degrees of time series correlation, which is measured using the autocorrelation function (ACF). i (t) and x i The correlation between (t+kk) and x, where kk is the lag. i ∈R n×1 (i=1,…,m), the autocorrelation formula for lag kk is:

[0012] ACF(kk)=S kk / S0 (2)

[0013]

[0014] When lag k > 0, if the ACF is within the given control range, the variable is considered to have no time-series correlation; otherwise, the variable is considered to have time-series correlation. For variables with strictly no time-series correlation, ACF = 0. Considering statistical significance, the control limit of ACF is set to 0.05, thus limiting the training dataset X. train ∈R n×m It is divided into dynamic subspace and non-dynamic subspace.

[0015] Step (2.2): Use the Jarque-Bera test to determine whether a single variable follows a Gaussian distribution, and then complete the division of Gaussian and non-Gaussian sub-blocks: for different variables x i ∈R n×1 (i = 1, 2, ..., m), with skewness SX and kurtosis KX, the variable strictly follows a Gaussian distribution when SX = 0 and KX = 3. A smaller JB statistic indicates a more Gaussian distribution. Furthermore, the JB statistic follows a χ² distribution with 2 degrees of freedom. 2 The distribution can be determined by finding the probability p that the statistic falls to the right tail of the current observation. Considering statistical significance, the confidence level is set to 0.05. When p > 0.05, the variable is considered to follow a Gaussian distribution; otherwise, it does not. Therefore, the training dataset X is... train ∈R n×M It is divided into Gaussian subspace and non-Gaussian subspace.

[0016]

[0017] Step (2.3): Based on the above division, divide the dataset X... train It is divided into four blocks: Gaussian non-dynamic block X1, non-Gaussian non-dynamic block X2, Gaussian dynamic block X3, and non-Gaussian dynamic block X4.

[0018] Step (3): For a Gaussian non-dynamic block X1, set the number of nearest neighbors K1 for local preservation, and use the NPE method to reduce the dimensionality of X1. Assume X1∈R n*m1 Finally, the dimensionality-reduced features are obtained. Dimensionality reduction mapping matrix P1. The specific steps are as follows:

[0019] Step (3.1): For the data x of X1 1i Find its K1 nearest neighbors to form a neighborhood N. i Using these neighborhoods to define the edges of the graph, we construct an adjacency matrix W, which represents the degree of connectivity between sample points. The elements in the adjacency matrix W are constructed as follows.

[0020]

[0021] Step (3.2): The scatter matrix S is defined as follows:

[0022]

[0023] Step (3.3): The NPE finds a mapping matrix P such that the representation of the sample points in the low-dimensional space can preserve the distance relationship between the original sample points. The objective function is constructed as follows:

[0024]

[0025] The objective function aims to minimize the distance between sample points in the neighborhood in a low-dimensional space.

[0026] Step (3.4): Construct the Lagrange function for optimization by introducing the Lagrange multiplier λ. Its form is:

[0027] L(P,λ)=J(P)+λ(P T PI) (12)

[0028] Where I is the identity matrix, ensuring that the sample variance remains unchanged after dimensionality reduction.

[0029] Step (3.5): By differentiating the optimization problem and setting it to zero, we obtain the eigenvalue problem. We then solve for P as the eigenvectors.

[0030] SP=λP (13)

[0031] Where S is the scatter matrix. Solving this eigenvalue problem yields the mapping matrix P.

[0032] Step (3.6): Map the original data X1 to a lower-dimensional space using the mapping matrix P:

[0033] Y1 = X1P (14) where Y1 is the feature data after dimensionality reduction.

[0034] Step (4): Construct the statistic GSD1, with the following formula:

[0035]

[0036] Where x1 is a data sample in X1, P is the dimension-reduced mapping matrix, and cov(Y1) is the covariance matrix of the dimension-reduced data Y1.

[0037] Step (5): For the non-Gaussian non-dynamic sub-block X2, the statistic GSD2 is obtained by following the NPE method above.

[0038] Step (6): For the Gaussian dynamic sub-block X3, the SFA method is used for feature extraction and dimensionality reduction to obtain the slow feature matrix W. d The specific steps are as follows:

[0039] Step (6.1): For dataset X3, calculate its derivative.

[0040]

[0041] Step (6.2): ​​Calculate the covariance matrix of the original dataset and the derivative dataset:

[0042] The covariance matrix of the derivative dataset is as follows:

[0043]

[0044] Original dataset covariance matrix:

[0045]

[0046] Step (6.3): Solve for the mapping matrix W of the slow features. d To extract slow features, the goal is to minimize the change in slow features:

[0047]

[0048] Constructing generalized eigenvalues:

[0049]

[0050] Solving this eigenvalue problem yields the slow eigenmap matrix W. d and the corresponding eigenvalue λ d .

[0051] Step (7): Construct the statistics GSD3 of the Gaussian dynamic sub-blocks, with the following formula:

[0052]

[0053] Where x3 is a data sample in X3, W d It is a slow feature matrix.

[0054] Step (8): For the non-Gaussian dynamic sub-block X4, feature extraction is still performed according to the above SFA method to construct the statistic GSD4.

[0055] Step (9): For X1, X2, X3, and X4, calculate the mutual information MM between each pair of them using a sliding time window. 12 MM 13 MM 14 MM 23 MM 24 MM 34 .

[0056] The specific steps are as follows:

[0057] Step (9.1): Taking blocks X1 and X2 as examples, the number of variables in X1 and X2 are m1 and m2 respectively. Set the sliding window length to KK. For any window position w in X1, and any variable x j Extract the variable data within the time window as: x 1,j (w)={x j (w),x j (w+1),…,x j (w+KK-1)},j=1,2,…,m1, similarly, extract the window position w in X2, and the variable x k The time window variable data in the middle is x 2,k (w).

[0058] Step (9.2): Calculate x 1,j (w) and x 2,k The mutual information of (w) is given by the following formula:

[0059]

[0060] Where p(x) 1,j ,x 2,k ) is x 1,j and x 2,k The joint probability density, p(x) 1,j ) and p(x 2,k ) are x 1,j and x 2,k The marginal probability density.

[0061] Step (9.3): For each window position w, calculate the mutual information values ​​of all possible variable pairs to form a mutual information matrix:

[0062]

[0063] Step (9.4): Summate all elements in the matrix to obtain the total mutual information MM within the w-th window. 12 (w), and then, based on the movement of the sliding window over the time series, the time series MM of the total mutual information is calculated sequentially. 12 The formula is as follows:

[0064]

[0065] MM 12 ={MM 12 (1),MM 12 (2),…,MM 12 (T)} (25)

[0066] Where T is the number of valid positions of the sliding window.

[0067] Step (9.5): Based on the above steps, calculate the other mutual information MMs. 13 MM 14 MM 23 MM 24 MM 34 .

[0068] Step (10): Using the LOF (Local Outlier Factor) method, consider the GSD statistics of each block. 1～4 And related information MM 12 MM 13 MM 14 MM 23 MM 24 MM 34 Establish a comprehensive statistical measure, LOF, and calculate its statistical threshold, LOFM. The specific steps are as follows:

[0069] Step (10.1): Construct the detection index matrix:

[0070] SS = [GSD1, GSD2, GSD3, GSD4, MM] 12 ,MM 13 ,MM 14 ,MM 23 ,MM 24 ,MM 34 (16)

[0071] Step (10.2): ss is a sample in SS. Construct its neighborhood space N(ss) based on its k nearest neighbor values. Define ss f Let d(ss, ss) be the f-th nearest neighbor of ss. f Let Fdis(s) be their Euclidean distance, and let Fdis(s) = d(ss, ss) k ), where ss k Let ss be the nearest neighbor with the furthest distance. Then Fdis(ss) f ) is ss f The Euclidean distance between it and its nearest and farthest neighbors. Let d(ss, ss) f ) and Fdis(ss f The larger value of ) is used as reach(ss, ss) f If the value of ) is given, then the locally reachable density and the LOF value are:

[0072]

[0073] Step (10.3) obtains the threshold LOFM of the LOF statistic by estimating kde using kernel density.

[0074] Steps (1) to (10) above constitute the offline modeling stage of the present invention, and steps (11) to (18) below constitute the online operation status monitoring process of the method of the present invention:

[0075] Step (11): Collect monitoring data x from m sensors t .

[0076] Step (12): Transfer data x t Based on the dataset X = [x1, x2, ..., x n ] T ∈R n*m The mean and standard deviation of the data are preprocessed to obtain standardized data x. t .

[0077] Step (13): Transfer data x t It is divided into four parts: Gaussian non-dynamic x t1 Non-Gaussian and non-dynamic x t2 Gaussian dynamics x t3 Non-Gaussian dynamics x t4 .

[0078] Step (16): Based on the offline modeling steps, construct data x respectively. t Statistics of each block GSD t1～4 .

[0079] Step (17): Construct data x according to the offline modeling steps. t Mutual information between each pair of blocks (MM)t12 MM t13 MM t14 MM t23 MM t24 MM t34 .

[0080] Step (18): Establish data x using the LOF method. t The comprehensive statistical measure LOF is compared with the statistical threshold LOFM to determine the operating status.

[0081] Compared with traditional methods, the advantages of the method of the present invention are as follows:

[0082] The dataset was divided into Gaussian dynamic, Gaussian non-dynamic, non-Gaussian dynamic, and non-Gaussian non-dynamic blocks. Appropriate feature extraction methods were applied to each type of variable, and mutual information between different blocks was incorporated, thereby improving the model's specificity and accuracy. Block processing better preserves the important information of each type of variable. Compared to directly building the entire model, it reduces interference between different variable characteristics, improves the accuracy and sensitivity of the operational status monitoring model, and makes the operational status monitoring results more accurate and reliable. Attached Figure Description

[0083] Figure 1 This is a flowchart of the method of the present invention.

[0084] Figure 2 This is a diagram showing the detection effect of fault 10 in an implementation example of the present invention. Detailed Implementation

[0085] The method of the present invention will be described in detail below with reference to the accompanying drawings and specific implementation examples.

[0086] This invention discloses a distributed operation status monitoring method based on local manifold structure and slow features, such as... Figure 1 The diagram shown illustrates the specific flowchart of this method, which not only captures dynamic and mutual information in process variables but also enables targeted modeling to improve the accuracy of operational status monitoring models. The following section will demonstrate the implementation process of this method and its advantages over existing methods using a specific industrial case study.

[0087] The Tennessee-Eastman (TE) process is a model based on actual chemical reaction processes and is widely used as a benchmark model for chemical process control and fault diagnosis. The TE process includes 41 process variables and 11 operational variables. It covers 21 different types of faults, providing a rich set of test scenarios for operational status monitoring. Tables 1 and 2 illustrate the different types of variables in the TE process, respectively.

[0088] Table 1: 22 continuous variables in the TE process

[0089]

[0090] Table 2: 11 manipulated variables in the TE process

[0091]

[0092] The implementation process of the method of this invention is as follows: using the data sampled by the TE process, an operational status monitoring model is established, including the following steps:

[0093] Step (1): Collect monitoring data of 33 variables at 960 sampling times under normal operating conditions to obtain the dataset X = [x1, x2, ..., x 960 ] T ∈R 960*33 Let R be the set of real numbers. The dataset is preprocessed using the z-score normalization method to obtain data X. train =[x train1 ,x train2 ,…,x train960 ] T ∈R 960*33 .

[0094] Step (2): Transfer dataset X train It is divided into four parts: Gaussian inactive X1 (variables 5, 6, 14, 15, 17, 29), non-Gaussian inactive X2 (variables 12, 26, 30, 33), Gaussian dynamic X3 (variables 1, 4, 8, 22, 23), and non-Gaussian dynamic X4 (variables 2, 3, 7, 9, 10, 11, 13, 16, 18, 19, 20, 21, 24, 25, 27, 28, 31, 32).

[0095] Step (3): For the Gaussian non-dynamic block X1, set the nearest neighbor count K1 = 50 for local preservation, and use the NPE method to reduce the dimensionality of X1. For X1∈R 960*6 We obtain the dimensionality-reduced feature dataset and the dimensionality-reduced mapping matrix, and then calculate the statistic GSD1.

[0096] Step (4): For non-Gaussian non-dynamic blocks, the NPE method above is still used to obtain the statistic GSD2.

[0097] Step (5): For Gaussian dynamic blocks, the SFA method is used to extract features and construct the Gaussian dynamic block statistic GSD3.

[0098] Step (6): For non-Gaussian dynamic blocks, perform dimensionality reduction using the SFA method described above and obtain the statistic GSD4.

[0099] Step (7): Calculate the mutual information MM between each pair of the four sub-blocks. 12 MM 13 MM 14 MM 23 MM 24 MM 34 .

[0100] Step (8): Using the LOF (Local Outlier Factor) method, consider the GSD statistics of each block. 1～4 And related information MM 12 MM 13 MM 14 MM 23 MM 24 MM 34 A comprehensive statistical measure, LOF, is established, and its statistical threshold, LOFM, is obtained through kde, thus completing the model construction.

[0101] The offline modeling phase is now complete; the next step is to analyze the online data.

[0102] Step (9): Collect monitoring data of 33 variables at 960 sampling times under test conditions to obtain the test dataset. The test dataset is based on X = [x1, x2, ..., x]. 960 ] T ∈R 960*33 Data preprocessing was performed using the mean and standard deviation to obtain the data.

[0103] Step (10): Divided into four sub-blocks: Gaussian non-dynamic (Variables 5, 6, 14, 15, 17, 29), non-Gaussian and non-dynamic (Variables 12, 26, 30, 33), Gaussian dynamics (Variables 1, 4, 8, 22, 23), Non-Gaussian dynamics (Variables 2, 3, 7, 9, 10, 11, 13, 16, 18, 19, 20, 21, 24, 25, 27, 28, 31, 32).

[0104] Step (11): Based on the offline modeling steps, use the SFA method to reduce the dimensionality of dynamic blocks and the NPE method to reduce the dimensionality of non-dynamic blocks. Then, calculate the GSD statistic for each block. t1～4 .

[0105] Step (12): Based on the offline modeling steps, calculate the mutual information (MM) between each pair of the four sub-blocks in the test dataset. t12 MM t13 MMt14 MM t23 MM t24 MM t34 .

[0106] Step (13): Consider the GSD statistics of each block t1～4 And related information MM t12 MM t13 MM t14 MM t23 MM t24 MM t34 The comprehensive statistical measure LOF is obtained and compared with its statistical threshold LOFM. If LOF > LOFM, the sample point is considered to have a fault; otherwise, no fault has occurred.

[0107] The detection results of this method and the classic method based on principal component analysis (PCA) are compared in Table 3.

[0108] Table 3: Detection results for fault 10

[0109]

[0110] Finally, the simulation results of the method of the present invention for fault 10 are shown in the figure. Figure 2 As can be seen, the method of the present invention has excellent detection performance and can effectively provide early warning of faults.

[0111] The above embodiments are only used to illustrate specific implementations of the present invention and are not intended to limit the invention. Therefore, any variations made in accordance with the form and principle of the present invention should be included within the scope of the present invention.

Claims

1. A distributed operation status monitoring method based on local manifold structure and slow characteristics, characterized in that, Includes the following steps: The implementation process of the offline modeling phase is as follows: Step (1): Data acquisition and data preprocessing, the specific implementation process is as follows: Step (1.1): Collect monitoring data from m sensors at n sampling times under normal operating conditions to obtain the dataset X = [x1, x2, ..., x...]. n ] T ∈R n*m , where n is the number of samples in the dataset, m is the number of sensors, and R is the set of real numbers. Step (1.2): Preprocess the data using the z-score normalization method, as shown in the following formula: Where x i It is a sample collected by a sensor, μ i It is the sample mean, σ i The standard deviation is the standard deviation. The dataset X is obtained after standardizing the data in each column. train ∈R n*m ; Step (2): According to X train The subspace is divided based on whether different variables in X exhibit dynamic characteristics and whether they follow a Gaussian distribution. train It is divided into four sub-blocks: Gaussian non-dynamic block X1, non-Gaussian non-dynamic block X2, Gaussian dynamic block X3, and non-Gaussian dynamic block X4. Step (3): For a Gaussian non-dynamic block X1, set the number of nearest neighbors K1 for local preservation, and use the NPE method to reduce the dimensionality of X1. Assume X1∈R n*m1 Finally, the dimensionality-reduced features are obtained. Dimensionality reduction mapping matrix P1. The specific steps are as follows: Step (3.1): For the data x of X1 1i Find its K1 nearest neighbors to form a neighborhood N. i Using these neighborhoods to define the edges of the graph, we construct an adjacency matrix W, which represents the degree of connectivity between sample points. The elements in the adjacency matrix W are constructed as follows. Step (3.2): The scatter matrix S is defined as follows: Step (3.3): The NPE finds a mapping matrix P such that the representation of sample points in the low-dimensional space can preserve the distance relationship between the original sample points. The objective function is constructed as follows: The objective function aims to minimize the distance between sample points in the neighborhood in a low-dimensional space. Step (3.4): Construct the Lagrange function for optimization by introducing the Lagrange multiplier λ. Its form is: L(P,λ)=J(P)+λ(P T PI) (5) Where I is the identity matrix, ensuring that the sample variance remains unchanged after dimensionality reduction. Step (3.5): By differentiating the optimization problem and setting it to zero, we obtain the eigenvalue problem. We then solve for P as the eigenvectors. SP=λP (6) Where S is the scatter matrix. Solving this eigenvalue problem yields the mapping matrix P. Step (3.6): Map the original data X1 to a lower-dimensional space using the mapping matrix P: Y1 = X1P (7) where Y1 is the feature data after dimensionality reduction. Step (4): Construct the statistic GSD1, with the following formula: Where x1 is a data sample in X1, P is the dimension-reduced mapping matrix, and cov(Y1) is the covariance matrix of the dimension-reduced data Y1. Step (5): For the non-Gaussian non-dynamic sub-block X2, the statistic GSD2 is obtained by following the NPE method above. Step (6): For the Gaussian dynamic sub-block X3, the SFA method is used for feature extraction and dimensionality reduction to obtain the slow feature matrix W. d The specific steps are as follows: Step (6.1): For dataset X3, calculate its derivative. Step (6.2): ​​Calculate the covariance matrix of the original dataset and the derivative dataset: The covariance matrix of the derivative dataset is as follows: Original dataset covariance matrix: Step (6.3): Solve for the mapping matrix W of the slow features. d To extract slow features, the goal is to minimize the change in slow features: Constructing generalized eigenvalues: Solving this eigenvalue problem yields the slow eigenmap matrix W. d and the corresponding eigenvalue λ d . Step (7): Construct the statistics GSD3 of the Gaussian dynamic sub-blocks, with the following formula: Where x3 is a data sample in X3, W d It is a slow feature matrix. Step (8): For the non-Gaussian dynamic sub-block X4, feature extraction is still performed according to the above SFA method to construct the statistic GSD4. Step (9): For X1, X2, X3, and X4, calculate the mutual information MM between each pair of them using a sliding time window. 12 MM 13 MM 14 MM 23 MM 24 MM 34 . Step (10): Using the LOF (Local Outlier Factor) method, consider the GSD (Gross Statistical Value) of each block. 1～4 And related information MM 12 MM 13 MM 14 MM 23 MM 24 MM 34 Establish a comprehensive statistical measure of LOF and calculate its statistical threshold LOFM. The specific steps are as follows: Step (10.1): Construct the detection index matrix: SS=[GSD1,GSD2,GSD3,GSD4,MM 12 ,MM 13 ,MM 14 ,MM 23 ,MM 24 ,MM 34 ] (15) Step (10.2): ss is a sample in SS. Construct its neighborhood space N(ss) based on its k nearest neighbor values. Define ss f Let d(ss, ss) be the f-th nearest neighbor of ss. f Let Fdis(s) be their Euclidean distance, and let Fdis(s) = d(ss, ss) k ), where ss k Let ss be the nearest neighbor with the furthest distance. Then Fdis(ss) f ) is ss f The Euclidean distance between it and its nearest and farthest neighbors. Let d(ss, ss) f ) and Fdis(ss f The larger value of ) is used as reach(ss, ss) f If the value of ) is given, then the locally reachable density and the LOF value are: Step (10.3) obtains the threshold LOFM of the LOF statistic by estimating kde using kernel density.

2. The distributed operation status monitoring method based on local manifold structure and slow characteristics according to claim 1, characterized in that, In step (2), the subspace is divided according to whether different variables are dynamic and whether they follow a Gaussian distribution. The specific implementation process is as follows: Step (2.1): The division of dynamic and non-dynamic subspaces is based on the temporal correlation of the variables themselves. That is, if a variable has temporal correlation, it is classified as a dynamic variable; otherwise, it is classified as a non-dynamic variable. Different types of x i ∈R n×1 (i = 1, 2, ..., m) exhibit varying degrees of time series correlation, which is measured using the autocorrelation function (ACF). i (t) and x i The correlation between (t+kk) and x, where kk is the lag. i ∈R n×1 (i=1,…,m), the autocorrelation formula for lag kk is: ACF(kk)=S kk / S0 (18) When lag k > 0, if the ACF is within the given control range, the variable is considered to have no time-series correlation; otherwise, the variable is considered to have time-series correlation. For variables with strictly no time-series correlation, ACF = 0. Considering statistical significance, the control limit of ACF is set to 0.05, thus limiting the training dataset X. train ∈R n×m It is divided into dynamic subspace and non-dynamic subspace. Step (2.2): Use the Jarque-Bera test to determine whether a single variable follows a Gaussian distribution, and then complete the division of Gaussian and non-Gaussian sub-blocks: for different variables x i ∈R n×1 (i = 1, 2, ..., m), with skewness SX and kurtosis KX, the variable strictly follows a Gaussian distribution when SX = 0 and KX = 3. A smaller JB statistic indicates a more Gaussian distribution. Furthermore, the JB statistic follows a χ² distribution with 2 degrees of freedom. 2 The distribution can be determined by finding the probability p that the statistic falls to the right tail of the current observation. Considering statistical significance, the confidence level is set to 0.

05. When p > 0.05, the variable is considered to follow a Gaussian distribution; otherwise, it does not. Therefore, the training dataset X is... train ∈R n×m It is divided into Gaussian subspace and non-Gaussian subspace. Step (2.3): Based on the above division, divide the dataset X... train It is divided into four blocks: Gaussian non-dynamic block X1, non-Gaussian non-dynamic block X2, Gaussian dynamic block X3, and non-Gaussian dynamic block X4.

3. The distributed operation status monitoring method based on local manifold structure and slow characteristics according to claim 1, characterized in that, In step (9), the mutual information between each sub-block is calculated using a sliding time window. The specific implementation process is as follows: Step (9.1): Taking blocks X1 and X2 as examples, the number of variables in X1 and X2 are m1 and m2 respectively. Set the sliding window length to KK. For any window position w in X1, and any variable x j Extract the variable data within the time window as: x 1,j (w)={x j (w),x j (w+1),…,x j (w+KK-1)},j=1,2,…,m1, similarly, extract the window position w in X2, and the variable x k The time window variable data in the middle is x 2,k (w). Step (9.2): Calculate x 1,j (w) and x 2,k The mutual information of (w) is given by the following formula: Where p(x) 1,j ,x 2,k ) is x 1,j and x 2,k The joint probability density, p(x) 1,j ) and p(x 2,k ) are x 1,j and x 2,k The marginal probability density. Step (9.3): For each window position w, calculate the mutual information values ​​of all possible variable pairs to form a mutual information matrix: Step (9.4): Summate all elements in the matrix to obtain the total mutual information MM within the w-th window. 12 (w), and then, based on the movement of the sliding window over the time series, the time series MM of the total mutual information is calculated sequentially. 12 The formula is as follows: MM 12 ={MM 12 (1),MM 12 (2),…,MM 12 (T)} (28) Where T is the number of valid positions of the sliding window. Step (9.5): Based on the above steps, calculate the other mutual information MMs. 13 MM 14 MM 23 MM 24 MM 34 .

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