On-the-fly learning semi-supervised soft sensor modeling method based on local label propagation

Abstract

The present invention relates to a real-time learning semi-supervised soft measurement modeling method based on local label propagation. The method fully extracts information from unlabeled samples through a local label propagation algorithm, establishes an online model about query data through an overall optimization real-time learning algorithm, and adds it to the local label propagation algorithm as a constraint item. For the collected query data, firstly, a local model is established according to the labeled historical data through an overall optimization real-time learning algorithm, then similar samples of the query data are selected from all the historical samples, and finally the output prediction value of the query data is calculated through a local label propagation algorithm based on local model constraints. The present invention can not only well handle the nonlinearity, time-varying and multicollinearity problems of industrial processes, but also effectively utilize a large amount of unlabeled historical data, improve the utilization rate of historical samples and the prediction accuracy of soft measurement models.

Description

Technical Field

[0001] The invention belongs to the technical field of industrial process detection, and relates to an industrial process soft measurement technology, and in particular to an instant learning semi-supervised soft measurement modeling method based on local label propagation. Background Art

[0002] In modern industrial production processes, many important quality variables (such as oil viscosity, components, etc.) are difficult to measure in real time, which has a great impact on chemical process control and optimization. Because the chemical production process has problems such as difficulty in on-site sampling, high cost of analytical instruments, and delayed analysis time, in the actual production process, it is often difficult to use online analytical instruments and offline testing to measure quality variables in real time, and it is impossible to form a closed-loop control of quality variables. Therefore, how to obtain quality variables in real time becomes the first problem to be solved in process control. As a result, soft measurement has entered the research field of process industrial control.

[0003] Commonly used data-driven soft sensor modeling methods include principal component regression (PCR), partial least squares regression (PLSR) and artificial neural network (ANN). The models established by the above soft sensor algorithms are offline models. After the model is established, it will not be adaptively adjusted with the changes in the generation process, and it cannot track the changes in the production status, resulting in a gradual decrease in prediction accuracy. Therefore, the automatic maintenance of soft sensor models has become the focus of algorithm research and improvement. Therefore, in order to adapt to the multimodal characteristics and time-varying characteristics of modern chemical production processes, various online modeling algorithms have been widely used in production process monitoring and quality variable prediction.

[0004] At present, the mainstream online soft sensor modeling algorithms include: moving window algorithm, recursive learning, time difference (TD) and just in time learning (JITL). Among them, the first three are to update the model based on time correlation, which are time adaptive algorithms; just in time learning algorithm is to update and maintain the model based on spatial correlation, which is a spatial adaptive algorithm. Compared with other algorithms, the advantage of just in time learning algorithm is that it can better adapt to the mutation phenomenon in the production process, and because the algorithm establishes a local model for each sample, it can also well describe the nonlinear relationship between process variables.

[0005] Although many modeling methods based on the just-in-time learning framework have been proposed recently and have achieved good results, most of them can only use labeled historical data. However, due to the low sampling frequency and long latency of quality variables, labeled samples often only account for a small part of all historical data. Therefore, using only a small number of labeled samples not only wastes a large number of unlabeled samples, but also may not accurately reflect the potential relationship between auxiliary variables and quality variables. As can be seen from the above, the model established using the traditional just-in-time learning algorithm has the problems of low sample utilization and poor prediction accuracy. Summary of the invention

[0006] In view of the above-mentioned problems of existing just-in-time learning technology, such as the inability to utilize unlabeled historical samples, the present invention provides a just-in-time learning semi-supervised soft measurement modeling method based on local label propagation (LLPJITL), which extends the just-in-time learning method to the semi-supervised field, can efficiently extract the information contained in unlabeled samples, and improves the model optimization efficiency and prediction accuracy.

[0007] In order to achieve the above object, the present invention provides a real-time learning semi-supervised soft sensor modeling method based on local label propagation, and the specific steps are as follows:

[0008] (I) Obtain the true values ​​of auxiliary variables and quality variables through industrial sensors and laboratory analysis, and obtain a total of n historical samples m1=m+1, where m is the number of auxiliary variables. l Historical samples are labeled samples The remaining n u Samples are unlabeled samples

[0009] (ii) Using the collected data as the initial training data set For the initial training data set According to formula (1), the training data set is standardized to make its mean 0 and variance 1. X std is the data obtained after data X is standardized. is the variable value obtained after the variable value is standardized, and formula (1) is expressed as:

[0010]

[0011] In the formula, the function mean(·) represents the calculation of the mean of each row of the matrix, and the function std(·) represents the calculation of the standard deviation of each row of the matrix;

[0012] (III) For newly collected query data According to formula (1), the normalization process is performed And add it to the historical database as an unlabeled sample

[0013] (IV) Based on the labeled samples in historical data First, the sample set is calculated by the improved collaborative representation learning algorithm (ICRL) The samples in The similarity weight Then, a local weighted ridge regression algorithm (WRR) is used to build a The local model of

[0014] (V) Historical Data In the example, select the spatial distance query data based on the Euclidean distance The most recent k1 historical samples are used as the spatial neighbor sample set (Include ); Then, the k2 temporal neighbor samples of all spatial neighbor samples are taken as the temporal neighbor sample set k=k1×k2; Finally, merge the spatial and temporal neighbor sample sets to obtain Double nearest neighbor sample k lp is the number of double neighbor samples;

[0015] (VI) Based on the nearest neighbor sample Compute query data through a label propagation algorithm based on local model constraints Output value

[0016] (VII) When the query data x is obtained through laboratory analysis or other methods q The actual output value y q When the sample [x q ,y q ] added to the training data set and re-standardize to expand the working range contained in the historical data; otherwise, maintain the training data set The space contained in remains unchanged.

[0017] Furthermore, in step (iv), the labeled samples are used The sample set is calculated by the improved collaborative representation algorithm The samples in The similarity weight The specific steps are:

[0018] Using sample sets An offline ridge regression model is established, and the optimization objective is:

[0019]

[0020] In the formula, is the ridge regression coefficient of the offline ridge regression model, λ0 is the regular term coefficient of the offline ridge regression model;

[0021] Solving the optimization objective, the analytical expression of the ridge regression coefficient θ0 of the offline ridge regression model is obtained as follows:

[0022] θ0=(X L X L T +λ0×I) -1 X L Y L (3)

[0023] Where, X L T For data X L The transpose of , I is the identity matrix;

[0024] According to the ridge regression coefficient θ0 of the offline ridge regression model, the weight matrix of each input variable is calculated by formula (4): Formula (4) is expressed as:

[0025]

[0026] Where θ0(1) is the first element of the ridge regression coefficient θ0, θ0(m) is the mth element of the ridge regression coefficient θ0, is the sum of the absolute values ​​of all elements of the model coefficient θ0;

[0027] Create query data With sample set The improved collaborative representation model between them has the following optimization objectives:

[0028]

[0029] In the formula, λ ICRL for The regularization coefficient of It is a diagonal matrix, and the diagonal elements are the Euclidean distances between each sample and the query data;

[0030] The synergy expression coefficient β is calculated by formula (6), which is expressed as:

[0031]

[0032] Using the synergistic representation coefficient β, the sample set is obtained through formula (7): and query data The similarity matrix W sample , formula (7) is expressed as:

[0033]

[0034] Where β(1) is the first element of the ridge regression coefficient β, β(n l ) is the nth ridge regression coefficient β l elements, S β is the sum of the absolute values ​​of all elements of the model coefficient β;

[0035] Furthermore, in step (iv), a local weighted ridge regression algorithm is used to establish a query data The local model and the local model coefficient θ JITL The specific steps are:

[0036] According to the sample set And its query data The similarity matrix W sample , the optimization objective of the weighted ridge regression algorithm is shown in formula (8):

[0037]

[0038] In the formula, λ WRR is the model coefficient The regularization coefficient of ;

[0039] The model coefficient θ of the local model is calculated by formula (9): JITL , formula (9) is expressed as:

[0040]

[0041] Furthermore, in step (six), the query data is calculated by a label propagation algorithm based on local model constraints Output value The specific steps are:

[0042] According to the selected double neighbor sample set and the local model coefficients θ JITL , the optimization goal of the label propagation algorithm based on local model constraints is:

[0043]

[0044] In the formula, for Pseudo labels of samples in , Y lp,i ,U i Similar sample sets The output value and weight of the i-th sample in the , the output value and weight of the unlabeled sample are both 0, is the similarity between the i-th and j-th samples, is the weighted ridge regression model coefficient, λ is the coefficient of the local model constraint;

[0045] All similar samples can be calculated by formula (11) The output prediction value of Output value Formula (11) is expressed as:

[0046]

[0047] Compared with the prior art, the present invention has the following beneficial effects:

[0048] The instant learning semi-supervised soft measurement modeling method based on local label propagation provided by the present invention integrates the label propagation instant learning algorithm and improves the label propagation algorithm. Specifically, on the one hand, the scope of label propagation is limited to the vicinity of the query sample, which improves the efficiency of label propagation and reduces the pseudo-connections between sample points; on the other hand, a local model is established through the instant learning algorithm and is integrated into the optimization target of the label propagation algorithm as a constraint item. Compared with other existing algorithms, the present invention combines instant learning with label propagation, which can not only make full use of unlabeled historical data to establish a soft measurement model, but also adapt to processes with nonlinear and time-varying characteristics, thereby improving model efficiency and prediction accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 It is a flow chart of the instant learning semi-supervised soft sensor modeling method based on local label propagation according to the present invention;

[0050] Figure 2 This is a schematic diagram of the sulfur recovery process (abbreviated as SRU) described in an embodiment of the present invention;

[0051] Figure 3 It is a real output curve diagram of the sulfur recovery process data described in the embodiment of the present invention;

[0052] Figure 4 A schematic diagram of the prediction deviation of sulfur recovery process data by the instant learning semi-supervised soft sensor modeling method based on local label propagation according to the present invention;

[0053] Figure 5 Schematic diagram of the prediction deviation of sulfur recovery process data using the existing local weighted partial least squares algorithm. DETAILED DESCRIPTION

[0054] The present invention is described in detail below by way of exemplary embodiments. However, it should be understood that elements, structures, and features in one embodiment may also be beneficially combined in other embodiments without further description.

[0055] The present invention aims at the time-varying and multimodal characteristics in industrial processes, as well as the problem of a small number of labeled samples commonly found in industrial data, and provides a real-time learning semi-supervised soft measurement modeling method based on local label propagation. A local model of query data is established through a real-time learning algorithm, overcoming the time-varying and multimodal problems; pseudo labels of unlabeled samples are calculated through a local label propagation algorithm, solving the problem of a small number of labeled samples. In addition, the present invention establishes a local model by overall optimizing the real-time learning algorithm, reducing the adjustable parameters in the algorithm, optimizing the modeling process, and improving the reliability of sample weights and the prediction accuracy of the soft measurement model. The real-time learning semi-supervised soft measurement modeling method based on local label propagation described in the present invention is described in detail below.

[0056] The embodiment of the present invention provides a real-time learning semi-supervised soft sensor modeling method based on local label propagation, and the specific steps are as follows:

[0057] (I) Obtain the true values ​​of auxiliary variables and quality variables through industrial sensors and laboratory analysis, and obtain a total of n historical samples m1=m+1, where m is the number of auxiliary variables. l Historical samples are labeled samples The remaining n u Samples are unlabeled samples

[0058] (ii) Using the collected data as the initial training data set For the initial training data set According to formula (1), the training data set is standardized to make its mean 0 and variance 1. X L is the data obtained after data X is standardized. is the variable value obtained after the variable value is standardized, and formula (1) is expressed as:

[0059]

[0060] In the formula, the function mean(·) represents the calculation of the mean of each row of the matrix, and the function std(·) represents the calculation of the standard deviation of each row of the matrix;

[0061] (III) For newly collected query data According to formula (1), the normalization process is performed And add it to the historical database as an unlabeled sample middle;

[0062] (IV) Based on the labeled samples in historical data First, the sample set is calculated by the improved collaborative representation algorithm The samples in x q The similarity weight Then, the local weighted ridge regression algorithm is used to build the query data The local model of

[0063] Specifically, using labeled samples The sample set is calculated by the improved collaborative representation algorithm The samples in The similarity weight The specific steps are:

[0064] Using sample sets An offline ridge regression model is established, and the optimization objective is:

[0065]

[0066] In the formula, is the ridge regression coefficient of the offline ridge regression model, λ0 is the regular term coefficient of the offline ridge regression model;

[0067] Solving the optimization objective, the analytical expression of the ridge regression coefficient θ0 of the offline ridge regression model is obtained as follows:

[0068] θ0=(X L X L T +λ0×I) -1 X L Y L (3)

[0069] Where, X L T For data X L The transpose of , I is the identity matrix;

[0070] According to the ridge regression coefficient θ0 of the offline ridge regression model, the weight matrix of each input variable is calculated by formula (4): Formula (4) is expressed as:

[0071]

[0072] Where θ0(1) is the first element of the ridge regression coefficient θ0, θ0(m) is the mth element of the ridge regression coefficient θ0, is the sum of the absolute values ​​of all elements of the model coefficient θ0;

[0073] Create query data With sample set The improved collaborative representation model between them has the following optimization objectives:

[0074]

[0075] In the formula, λ ICRL for The regularization coefficient of It is a diagonal matrix, and the diagonal elements are the Euclidean distances between each sample and the query data;

[0076] The synergy expression coefficient β is calculated by formula (6), which is expressed as:

[0077] β=(X L T W var X L +λ ICRL D) -1 X L T W var x q (6)

[0078] Using the synergistic representation coefficient β, the sample set is obtained through formula (7): and query data The similarity matrix W sample , formula (7) is expressed as:

[0079]

[0080] Where β(1) is the first element of the ridge regression coefficient β, β(n l ) is the nth ridge regression coefficient β l elements, S β is the sum of the absolute values ​​of all elements of the model coefficient β;

[0081] The local weighted ridge regression algorithm is used to build The local model and the local model coefficient θ JITL The specific steps are:

[0082] According to the sample set And its query data The similarity matrix W sample , the optimization objective of the weighted ridge regression algorithm is shown in formula (8):

[0083]

[0084] In the formula, λ WRR for The regularization coefficient of ;

[0085] The model coefficient θ of the local model is calculated by formula (9): JITL , formula (9) is expressed as:

[0086]

[0087] (V) Historical Data In the example, select the spatial distance query data based on the Euclidean distance The most recent k1 historical samples are used as the spatial neighbor sample set (Include ); Then, the k2 temporal neighbor samples of all spatial neighbor samples are taken as the temporal neighbor sample set k=k1×k2; Finally, merge the spatial and temporal neighbor sample sets to obtain Double nearest neighbor sample k lp is the number of double neighbor samples;

[0088] (VI) Based on the nearest neighbor sample Compute query data through a label propagation algorithm based on local model constraints Output value The specific steps are:

[0089] According to the selected double neighbor sample set and the local model coefficients θ JITL , the optimization goal of the label propagation algorithm based on local model constraints is:

[0090]

[0091] In the formula, for Pseudo labels of samples in , Y lp,i ,U i Similar sample sets The output value and weight of the i-th sample in the , the output value and weight of the unlabeled sample are both 0, is the similarity between the i-th and j-th samples, is the weighted ridge regression model coefficient, λ is the coefficient of the local model constraint;

[0092] All similar samples can be calculated by formula (11) The output prediction value of Output value Formula (11) is expressed as:

[0093]

[0094] (VII) When the query data x is obtained through laboratory analysis or other methods q The actual output value y q When the sample [x q ,y q ] added to the training data set and re-standardize to expand the working range contained in the historical data; otherwise, maintain the training data set The space contained in remains unchanged.

[0095] The above method of the embodiment of the present invention efficiently obtains the information contained in the unlabeled historical data through the local label propagation algorithm, and establishes a local model based on the labeled historical samples through the overall optimization instant learning algorithm to obtain the information in the labeled samples, and finally fuses the two types of information. For the collected query data, firstly, an online model is established based on the labeled historical data through the overall optimization instant learning algorithm, and then similar samples of the query data are selected from all historical samples. Finally, based on the selected similar samples, the output prediction value of the query data is calculated through the local label propagation algorithm based on the online model constraints. The present invention can not only handle the nonlinearity, time-varying and multicollinearity problems of industrial processes well, but also effectively utilize a large amount of unlabeled historical data, improve the utilization rate of historical samples and the prediction accuracy of soft measurement models.

[0096] In order to illustrate the effect of the instant learning soft sensor modeling method based on local label propagation of the present invention, the present invention is further described below in conjunction with specific embodiments.

[0097] Example: The process data of sulfur recovery is used as an example for explanation.

[0098] Sulfur recovery is an important refinery process unit (SRU). Before the acid gas stream is released into the atmosphere, it removes environmental pollutants and recovers the sulfur element contained in it. In order to better remove sulfides, the concentration ratio of hydrogen sulfide to sulfur dioxide must be controlled at 1:2. This can be achieved by establishing a closed-loop control between the concentrations of hydrogen sulfide (H2S) and sulfur dioxide (SO2) and the air supply ratio, but this requires real-time monitoring of the concentrations of hydrogen sulfide and sulfur dioxide. In addition, since these two acid gases are highly corrosive to hardware instruments, the instruments need to be replaced and maintained frequently, which greatly increases production costs. Therefore, the concentrations of SO2 and H2S can be predicted in real time through a soft sensing model. This article takes the prediction of SO2 concentration as an example. Table 1 gives an explanation of the five auxiliary variables and their positions in the process. Figure 1 The data set comes from an actual industrial production process. The number of historical samples is 10072. They are arranged by sampling time. The first 7001 are used as test samples, and the remaining 3071 are test samples. The actual output curve of the test set is shown in Figure 2 In addition, considering the low sampling efficiency and long delay of quality variables in the actual production process, it is assumed that only one eighth of all historical samples have labels (including both input data and output data), and the other historical samples are unlabeled samples (including only input data).

[0099] Table 1

[0100] Auxiliary variables Variable Description <![CDATA[X1]]> MEA gas flow <![CDATA[X2]]> MEA zone first air flow <![CDATA[X3]]> Second air flow rate in MEA zone <![CDATA[X4]]> SWS Gas Flow <![CDATA[X5]]> SWS zone air flow

[0101] Next, the specific steps of the present invention are described in conjunction with the sulfur recovery production process:

[0102] 1. Use the collected data as a training data set and preprocess it;

[0103] First, all samples were preprocessed and abnormal samples were deleted, resulting in a total of 7001 historical data. Among them, the first 875 are labeled samples The remaining 6126 are unlabeled samples. Then, considering the dynamic characteristics of the process, all samples are dimensionally expanded according to the following formula, and the dimension of the expanded samples is 20; finally, the final training data set is obtained by standardization. but:

[0104]

[0105] u(t)=(x1(t),x2(t),x3(t),x4(t),x5(t))

[0106] In the formula, represents the predicted value of sulfur dioxide concentration by the soft sensor model, f SRU (·) represents the potential relationship between sulfur dioxide concentration and X1~X5.

[0107] 2. Collect new data and standardize it;

[0108] For newly collected query data Standardize according to the training set standard And add it to the historical database as an unlabeled sample middle;

[0109] 3. Build a local model based on the labeled historical data;

[0110] Based on the labeled samples in historical data First, the sample set is calculated by the improved collaborative representation algorithm The samples in Similarity weight Then, the query data x is constructed by using the local weighted ridge regression algorithm. q The local model of

[0111] 4. Select the double nearest neighbor samples of the query data from all historical samples;

[0112] In historical data In the example, select the spatial distance query data based on the Euclidean distance The most recent k1 historical samples are used as the spatial neighbor sample set (Include ); Then, the k2 temporal neighbor samples of all spatial neighbor samples are taken as the temporal neighbor sample set k=k1×k2; Finally, merge the spatial and temporal neighbor sample sets to obtain Double nearest neighbor sample k lp is the number of double neighbor samples;

[0113] 5. Calculate the output value of the query data through the label propagation algorithm based on local model constraints;

[0114] Based on the nearest neighbor sample Compute query data through a label propagation algorithm based on local model constraints Output value When the true value of the output variable y is obtained q When the sample [x q ;y q ]Add to the training dataset and re-standardize it;

[0115] The prediction deviation values ​​of the output variables of sulfur recovery data by the method of the present invention (abbreviated as: LLPJ I TL) and the traditional local weighted partial least squares (abbreviated as: LWPLS) algorithm are as follows: Figure 3 , Figure 4 As shown. Figure 3 , Figure 4 It can be seen that the method of the present invention has higher prediction accuracy than the traditional method.

[0116] The above embodiments are used to explain the present invention rather than to limit the present invention. Any modification and change made to the present invention within the spirit of the present invention and the protection scope of the claims shall fall within the protection scope of the present invention.

Claims

1. A real-time learning semi-supervised soft sensor modeling method based on local label propagation, characterized in that: The specific steps are: (I) Obtain the true values ​​of auxiliary variables and quality variables through industrial sensors and laboratory analysis, and obtain a total of n historical samples m is the number of auxiliary variables, among which the first n l Historical samples are labeled samples The remaining n u Samples are unlabeled samples (ii) Using the collected data as the initial training data set For the initial training data set According to formula (1), the training data set is standardized to make its mean 0 and variance 1. X std is the data obtained after data X is standardized. is the variable value obtained after the variable value is standardized, and formula (1) is expressed as: In the formula, the function mean(·) represents the calculation of the mean of each row of the matrix, and the function std(·) represents the calculation of the standard deviation of each row of the matrix; (III) For newly collected query data According to formula (1), the normalization process is And add it to the historical database as an unlabeled sample middle; (IV) Based on the labeled samples in historical data First, the sample set is calculated by the improved collaborative representation algorithm The samples in The similarity weight Then, the local weighted ridge regression algorithm is used to build the query data The local model of (V) Historical Data In the example, select the spatial distance query data based on the Euclidean distance The most recent k1 historical samples are used as the spatial neighbor sample set Then, the k2 temporal neighbor samples of all spatial neighbor samples are taken as the temporal neighbor sample set Finally, merge the spatial and temporal neighbor sample sets to obtain Double nearest neighbor sample k lp is the number of double neighbor samples; (VI) Based on the nearest neighbor sample Compute query data using a label propagation algorithm based on local model constraints Output value (VII) When the query data x is obtained through laboratory analysis q The actual output value y q When the sample [x q ,y q ] added to the training data set and re-standardize to expand the working range included in the historical data; Otherwise, keep the training dataset The space contained in remains unchanged.

2. The instant learning semi-supervised soft sensor modeling method based on local label propagation according to claim 1 is characterized in that: In step (iv), using the labeled samples The sample set is calculated by the improved collaborative representation algorithm The samples in The similarity weight The specific steps are: Using sample sets An offline ridge regression model is established, and the optimization objective is: In the formula, is the ridge regression coefficient of the offline ridge regression model, λ0 is the regular term coefficient of the offline ridge regression model; Solving the optimization objective, the analytical expression of the ridge regression coefficient W0 of the offline ridge regression model is obtained as follows: θ0=(X L X L T +λ0×I) -1 X L Y L (3) In the formula, X L T For data X L The transpose of , I is the identity matrix; According to the ridge regression coefficient θ0 of the offline ridge regression model, the weight matrix of each input variable is calculated by formula (4): Formula (4) is expressed as: Where θ0(1) is the first element of the ridge regression coefficient θ0, θ0(m) is the mth element of the ridge regression coefficient θ0, is the sum of the absolute values ​​of all elements of the model coefficient θ0; Create query data With sample set The improved collaborative representation model between them has the following optimization objectives: In the formula, λ ICRL for The regularization coefficient of It is a diagonal matrix, and the diagonal elements are the Euclidean distances between each sample and the query data; The synergy expression coefficient β is calculated by formula (6), which is expressed as: Using the synergistic representation coefficient β, the sample set is obtained through formula (7): and query data The similarity matrix W sample , formula (7) is expressed as: Where β(1) is the first element of the ridge regression coefficient β, β(n l ) is the nth ridge regression coefficient β l elements, S β is the sum of the absolute values ​​of all elements of the model coefficient β.

3. The instant learning semi-supervised soft sensor modeling method based on local label propagation as claimed in claim 2, characterized in that: In step (iv), a local weighted ridge regression algorithm is used to establish the query data The local model and the local model coefficient θ JITL The specific steps are: According to the sample set And its query data The similarity matrix W sample , the optimization objective of the weighted ridge regression algorithm is shown in formula (8): In the formula, λ WRR is the model coefficient The regularization term coefficient of ; The model coefficient θ of the local model is calculated by formula (9): JITL , formula (9) is expressed as:

4. The instant learning semi-supervised soft sensor modeling method based on local label propagation as claimed in claim 3, characterized in that: In step (six), the query data is calculated using a label propagation algorithm based on local model constraints Output value The specific steps are: According to the selected double nearest neighbor sample set and the local model coefficients θ JITL , the optimization goal of the label propagation algorithm based on local model constraints is: In the formula, for Pseudo labels of samples in , Y lp,i ,U i Similar sample sets The output value and weight of the i-th sample in the , the output value and weight of the unlabeled sample are both 0, is the similarity between the i-th and j-th samples, is the weighted ridge regression model coefficient, λ is the coefficient of the local model constraint; All similar samples can be calculated by formula (11) The output prediction value of Output value Formula (11) is expressed as:

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