ScRNA-Seq data interpolation method based on graph convolutional neural network and statistical model constraint

Abstract

The invention discloses an scRNA-seq data interpolation method based on a graph convolutional neural network and statistical model constraint. The scRNA-seq data interpolation method comprises the following steps: 1) data preprocessing; 2) graph information construction and pre-label acquisition; 3) carrying out GCN-based preliminary interpolation; and 4) carrying out statistical model constraint based on negative binomial / zero-expansion negative binomial distribution. The method is high in interpolation precision and practical value and good in downstream analysis effect, and a new thought and a new method are provided for interpolation of scRNA-seq data.

Description

Technical Field

[0001] The present invention relates to the field of bioinformatics, and mainly adopts a deep learning method to interpolate scRNA-seq data, specifically a scRNA-seq data interpolation method based on a graph convolutional neural network and statistical model constraints. Technical Background

[0002] Transcriptome sequencing is an important means of studying the transcriptome. It is of great help in gene expression analysis, exploration of new transcripts, etc., and has been widely used in basic research and drug development.

[0003] Traditional batch RNA-seq sequencing technology can only calculate the average expression level of each gene in a cell population, and cannot reflect the heterogeneity of gene expression in a single cell. However, major diseases faced by all mankind, such as cancer, are highly heterogeneous. There is heterogeneity not only between tissues, but also between cells. Discovering this heterogeneity plays an important role and significance in promoting personalized medicine. In order to overcome the shortcomings of batch RNA-seq sequencing technology, single-cell sequencing (scRNA-seq) technology was developed and first published in 2009. Unfortunately, due to the imperfect and immature technology, scRNA-seq data usually have a lot of noise, the most common of which is dropout events, that is, missing values, and is usually represented by 0 values ​​in the gene-cell matrix of scRNA-seq data. However, there are two cases of these 0 values. One is the real 0 value that the gene is not expressed, and the other is the false 0 value caused by defects in sequencing technology. These false 0 values ​​will cause errors or deviations in downstream analysis (such as cell clustering, differential expression analysis, and cell trajectory inference, etc.), which will weaken the application of scRNA-seq data in biological and medical research. Therefore, how to identify dropout events and interpolate appropriate data where dropout occurs is an urgent need for scRNA-seq data analysis.

[0004] There are many methods to interpolate scRNA-seq data for the dropout problem. Linear model-based methods usually ignore the nonlinear relationships in the data, which cannot reflect the randomness of gene expression, which is ubiquitous in life processes. Deep learning-based methods can effectively learn nonlinear relationships and some potential features in scRNA-seq data, but the interpretability is poor. The scRNA-seq data optimization model based on graph neural networks adds graph information (such as extracted cell-cell or gene-gene relationships), which greatly enhances the interpretability of deep learning methods for interpolating scRNA-seq data. However, graph neural networks still have some defects. For example, using K nearest neighbors to construct graph information between cells will cause the interpolation results to be overly smooth. Summary of the invention

[0005] In view of the shortcomings of existing methods for interpolating scRNA-seq data, the present invention proposes a method for interpolating scRNA-seq data based on graph convolutional neural networks and statistical model constraints. This method has high interpolation accuracy and good downstream analysis effect, and provides new ideas and methods for interpolating scRNA-seq data.

[0006] The technical solution for achieving the purpose of the present invention is as follows:

[0007] 1) Data preprocessing:

[0008] The scRNA-seq data is used as input, and the input scRNA-seq data is preprocessed for data filtering and quality control: first, only genes expressed in more than 1% of cells and cells expressed in more than 1% of genes are retained; second, if the number of retained genes is too large, the retained genes are sorted by standard deviation, and the top 1000 genes with the largest variance are selected for research, thus obtaining the scRNA-seq data X for input into the model;

[0009] 2) Graph information construction and pre-label acquisition:

[0010] Based on GCN, preliminary interpolation is performed, so the graph information between cell samples, that is, the adjacency matrix A between cells, is required as input. The K nearest neighbor algorithm, that is, KNN, is used to construct the graph information. In order to avoid the noise in X and provide more accurate information for the subsequent construction of the adjacency matrix, PCA dimensionality reduction is first performed on X:

[0011] X d = PCA(X) (1)

[0012] Then use the K nearest neighbor algorithm to find X d Construct the adjacency matrix:

[0013] A=KNN(X d ) (2)

[0014] The top-k parameter of KNN is set to 5, and then d Perform pre-clustering to obtain the pre-label of each sample point:

[0015] y p =KMeans(X d ) (3)

[0016] At this point, all the input data required for GCN training is prepared;

[0017] 3) Preliminary interpolation based on GCN:

[0018] In order to identify dropout events in scRNA-seq data and interpolate them, a GCN-based model was trained. Compared with traditional neural network models, the main advantage of GCN is that it makes full use of the graph structure information between sample points, making the estimation process more interpretable and having better interpolation results. First, the adjacency matrix A is processed to obtain a normalized Laplace matrix:

[0019]

[0020] in I represents the identity matrix of the same size as A, and Finally, input X and the normalized Laplacian adjacency matrix into GCN and obtain preliminary interpolation results:

[0021]

[0022] Where W i represents the weight parameter of the i-th layer of GCN, σ represents the activation function of each layer, and in formula (5), we use Rather than The reason is that in the iteration of training GCN, the initial normalized Laplacian matrix is ​​not always used. Instead, as the iteration proceeds, the better data X′ output by the model is used to construct a better normalized Laplacian adjacency matrix in the order of formulas (1), (2), and (4). For subsequent iterative training,

[0023] Because GCN is still not accurate enough in reconstructing and denoising scRNA-seq data, we added a design based on GCN that enables GCN to accurately identify dropout events and make more reasonable interpolation:

[0024] 3.1) In order to make the GCN model not make changes to the places with expression values ​​in X, a separate mean square error loss function is designed for the places with expression values ​​in X, and the places with expression values ​​in X are recorded as 1 to obtain a labeling matrix e of the same size as X:

[0025]

[0026] And use the loss function L1 to guide the training of the corresponding weights:

[0027] L1=MSE(X·e,X′·e) (7) 3.2) Since the dropout event only occurs where the expression value is zero in X, in order to make the initial interpolation of the GCN model be performed where the expression value is 0, a similar strategy as step 3.1) is adopted to mark the expression value 0 in X with 1 to obtain a marking matrix of the same size as X

[0028]

[0029] And use the loss function L2 to guide the training of the corresponding weights:

[0030]

[0031] Among them, α is a tolerance factor specially designed for the loss function L2, and its range is [0,1). The larger α is, the larger the interpolation value is where the expression value is 0. σ is a Gaussian kernel function parameter, which can be calculated by the number of cells with expression values ​​0 in X and the initialized α. The following are the steps to calculate the parameter σ: Assuming that the size of the existing X to be interpolated is n*m, and there are p 0 values ​​in X, then the proportion of 0 values ​​in X is: γ=p / (n*m), then:

[0032] min`error=(p*λ*a) 2 (10)

[0033] max`error=(p*λ*b) 2 (11) in λ indicates the number of zero values ​​that need to be interpolated. Since scRNA-seq data is sparse, λ is set to a small value in order not to destroy the inherent laws of the data. In addition, a and b indicate the range of values ​​that need to be interpolated at zero, which can also be expressed as [a, b]. a and b are hyperparameters that can be set by yourself.

[0034] Then, the X-specific σ value is calculated according to the following equation and algorithm flow:

[0035]

[0036]

[0037] 3.3) The data processed by GCN will have the phenomenon of overfitting. In order to overcome this problem, a classifier is added to the bottleneck layer of the GCN in the shape of the autoencoder. The classifier can alleviate the overfitting phenomenon through the preset hyperparameter K, and by presetting a classification loss function L3, the final interpolation result and downstream analysis can obtain good performance. The loss function L3 is used to guide the training of the corresponding weights:

[0038] L3=CrossEntropy(y p ,y) (14) where y p is the pre-label of each sample obtained in step 2), y is the multi-classification value obtained by the classifier, and the classifier is composed of a fully connected neural network, whose input Z is the output of the bottleneck layer of the autoencoder:

[0039]

[0040] Get y p The number of clusters K of KMeans is actually unknown, so a list K = [n-1, n, n+1] is set up. p When , the three cluster numbers in K are used respectively. Similarly, in order to p Correspondingly, the number of neurons in the output layer of the classifier is also changed to the number in K. This strategy means that the GCN model needs to be trained three times with different numbers of clusters. The final estimated data given by GCN is the average of these three estimates, where n is a user-defined hyperparameter.

[0041] After the above steps 3.1), 3.2) and 3.3), the final loss function L of GCN is obtained:

[0042] L=β*L1+(1-β)*(L2+L3) (16) β is a parameter set by yourself;

[0043] 4) Statistical model constraints based on negative binomial or zero-inflated negative binomial distribution:

[0044] After the preliminary interpolation in step 3), in order to prevent GCN from incorrectly interpolating the true zero value, the negative binomial or zero-inflated negative binomial distribution statistical model is used to further constrain the interpolation results:

[0045]

[0046] Formula (17) is the probability mass equation of the negative binomial distribution, where X is the scRNA-seq data, also called the observed variable in formula (17), ф and μ are the parameters of the negative binomial distribution. On the basis of the NB distribution, the probability mass equation of the ZINB distribution can be obtained by adding the zero inflation factor:

[0047]

[0048] Through the negative binomial or zero-inflated negative binomial distribution statistical tools, the probability of the expression value appearing in X under the corresponding distribution can be calculated. The specific estimation method used is the EM algorithm:

[0049]

[0050]

[0051] The statistical model obtains parameters After that, substitute X into the statistical model to obtain a probability matrix of the same size as X:

[0052]

[0053] Finally, use Through a weighted equation, the preliminary interpolation data X′ of the GCN model is constrained with reference to X to obtain the final interpolation data

[0054]

[0055] At this point, the entire interpolation process is completed.

[0056] Advantages of this technical solution:

[0057] (1) This technical solution uses GCN as the basic model, which can utilize the similarity information between samples to better interpolate scRNA-seq data;

[0058] (2) According to the characteristics of dropout in scRNA-seq data, the GCN framework based on this technical solution designs unique components for the interpolation of scRNA-seq data, such as a loss function that can improve the accuracy of interpolation and a classifier that effectively prevents model overfitting;

[0059] (3) In order to further distinguish dropout events from true zero values ​​and allow the graph convolutional neural network-based model to perform interpolation only at the dropout sites as much as possible, this technical solution also introduces a statistical model based on negative binomial or zero-inflated negative binomial distribution for final constraints, making the interpolation results more interpretable.

[0060] This method has high interpolation accuracy and practical value, and good downstream analysis effect, providing new ideas and methods for the interpolation of scRNA-seq data. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 Schematic diagram of the experimental results of dimensionality reduction visualization based on T-SNE algorithm for each comparison method in the embodiment;

[0062] Figure 2 Schematic diagram of the experimental results of the mean square error comparison between the estimated data and the real data of each comparison method in the embodiment;

[0063] Figure 3 It is a schematic diagram of comparative experimental results of interpolation ratios of the comparison methods in the embodiment on the real 0 value and the dropout 0 value. DETAILED DESCRIPTION

[0064] The invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments, but the invention is not limited thereto.

[0065] Example:

[0066] This experiment was conducted on a computer (Intel(R) Core(TM) i7-12700 2.10GHz CPU, 16G memory), the operating system was Windows 10, and the compilation and running tool was PyCharm2023.

[0067] This example uses a simulated "true data" (True data, i.e., no dropout at all) generated by the Splatter toolkit in R language and the data (Raw data) obtained by performing 50% random dropout on the non-zero entries of the True data as the test data set. The True data contains 500 single-cell samples, each of which contains 1,000 genes, and these 500 samples are divided into 5 subtypes.

[0068] In order to verify the performance of this example, DCA, MAGIC, scImpute, SAVER, scGNN, AutoClass, AGImpute|, DeepImpute, ALRA and CL-Impute are used as comparison methods, and experiments are conducted on the Raw data dataset. The parameters of these ten representative methods all use their default parameters. The parameters used in this example are shown in Table 1.

[0069] Table 1 Parameters used in this example

[0070]

[0071] The method for scRNA-seq data interpolation based on graph convolutional neural network and statistical model constraints includes the following steps:

[0072] 1) Data preprocessing:

[0073] The scRNA-seq data is used as input, and the input scRNA-seq data is preprocessed for data filtering and quality control: first, only genes expressed in more than 1% of cells and cells expressed in more than 1% of genes are retained; second, if the number of retained genes is too large, the retained genes are sorted by standard deviation, and the top 1000 genes with the largest variance are selected for research, thus obtaining the scRNA-seq data X for input into the model;

[0074] 2) Graph information construction and pre-label acquisition:

[0075] Based on GCN, preliminary interpolation is performed, so the graph information between cell samples, that is, the adjacency matrix A between cells, is required as input. The K nearest neighbor algorithm, that is, KNN, is used to construct the graph information. In order to avoid some noise in X and provide more accurate information for the subsequent construction of the adjacency matrix, PCA dimensionality reduction is first performed on X:

[0076] X d = PCA(X) (1)

[0077] Then use the K nearest neighbor algorithm to find X d Construct the adjacency matrix:

[0078] A=KNN(X d ) (2)

[0079] The top-k parameter of KNN is set to 5, and then d Perform pre-clustering to obtain the pre-label of each sample point:

[0080] y p =KMeans(X d ) (3)

[0081] At this point, all the input data required for GCN training is prepared;

[0082] 3) Preliminary interpolation based on GCN:

[0083] In order to identify dropout events in scRNA-seq data and interpolate them, a GCN-based model was trained. First, the adjacency matrix A was processed to obtain a normalized Laplacian matrix:

[0084]

[0085] in I represents the identity matrix of the same size as A, and Finally, input X and the normalized Laplacian adjacency matrix into GCN and obtain preliminary interpolation results:

[0086]

[0087] Where W i represents the weight parameter of the i-th layer of GCN, σ represents the activation function of each layer, and in formula (5), we use Rather than The reason is that in the iteration of training GCN, the initial normalized Laplacian matrix is ​​not always used. Instead, as the iteration proceeds, the better data X′ output by the model is used to construct a better normalized Laplacian adjacency matrix in the order of formulas (1), (2), and (4). For subsequent iterative training,

[0088] Because GCN is still not accurate enough in reconstructing and denoising scRNA-seq data, we added a design based on GCN that enables GCN to accurately identify dropout events and make more reasonable interpolation:

[0089] 3.1) In order to make the GCN model not make changes to the places with expression values ​​in X, a separate mean square error loss function is designed for the places with expression values ​​in X, and the places with expression values ​​in X are recorded as 1 to obtain a labeling matrix e of the same size as X:

[0090]

[0091] And use the loss function L1 to guide the training of the corresponding weights:

[0092] L1=MSE(X·e,X′·e) (7) 3.2) Since the dropout event only occurs where the expression value is zero in X, in order to make the initial interpolation of the GCN model be performed where the expression value is 0, a similar strategy as step 3.1) is adopted to mark the expression value 0 in X with 1 to obtain a marking matrix of the same size as X

[0093]

[0094] And use the loss function L2 to guide the training of the corresponding weights:

[0095]

[0096] Among them, α is a tolerance factor specially designed for the loss function L2, and its range is [0,1). In this example, the value of α is set to 0.3. The larger α is, the larger the interpolation value is where the expression value is 0. σ is a Gaussian kernel function parameter, which can be calculated by the number of cells with expression values ​​0 in X and the initialized α. The following are the steps to calculate the parameter σ: Assuming that the size of the existing X to be interpolated is n*m, and there are p 0 values ​​in X, then the proportion of 0 values ​​in X is: γ=p / (n*m), then:

[0097] min`error=(p*λ*a) 2 (10)

[0098] max`error=(p*λ*b) 2 (11)

[0099] in λ indicates the number of zero values ​​that need to be interpolated. Since scRNA-seq data is sparse, λ is set to a small value in order not to destroy the inherent laws of the data. In addition, a and b indicate the range of values ​​that need to be interpolated at zero, which can also be expressed as [a, b]. a and b are hyperparameters that can be set by yourself.

[0100] Then, the X-specific σ value is calculated according to the following equation and algorithm flow:

[0101]

[0102]

[0103] 3.3) The data processed by GCN will have the phenomenon of overfitting. In order to overcome this problem, a classifier is added to the bottleneck layer of the GCN in the shape of the autoencoder. The classifier can alleviate the overfitting phenomenon through the preset hyperparameter K, and by presetting a classification loss function L3, the final interpolation result and downstream analysis can obtain good performance. The loss function L3 is used to guide the training of the corresponding weights:

[0104] L3=CrossEntropy(y p ,y) (14)

[0105] where y p is the pre-label of each sample obtained in step 2), y is the multi-classification value obtained by the classifier, and the classifier is composed of a fully connected neural network, whose input Z is the output of the bottleneck layer of the autoencoder:

[0106]

[0107] Get y pThe number of clusters K of KMeans is actually unknown, so a list K = [n-1, n, n+1] is set up. p When , the three cluster numbers in K are used respectively. Similarly, in order to p Correspondingly, the number of neurons in the output layer of the classifier is also changed to the number in K. This strategy means that the GCN model needs to be trained three times with different numbers of clusters. The final estimated data given by GCN is the average of these three estimates, where n is a user-defined hyperparameter.

[0108] After the above steps 3.1), 3.2) and 3.3), the final loss function L of GCN is obtained:

[0109] L=β*L1+(1-β)*(L2+L3) (16)

[0110] The parameter β set in this example is 0.9;

[0111] The process of obtaining the preliminary interpolation result X′ of Raw data from steps 1), 2), and 3) is shown in Algorithm 2:

[0112]

[0113] 4) Statistical model constraints based on negative binomial or zero-inflated negative binomial distribution:

[0114] After the preliminary interpolation in step 3), in order to prevent GCN from incorrectly interpolating the true zero value, the negative binomial or zero-inflated negative binomial distribution statistical model is used to further constrain the interpolation results:

[0115]

[0116] Formula (17) is the probability mass equation of the negative binomial distribution, where X is the scRNA-seq data, also called the observed variable in formula (17), ф and μ are the parameters of the negative binomial distribution. On the basis of the NB distribution, the probability mass equation of the ZINB distribution can be obtained by adding the zero inflation factor:

[0117]

[0118] Through the negative binomial or zero-inflated negative binomial distribution statistical tools, the probability of the expression value appearing in X under the corresponding distribution can be calculated. The specific estimation method used is the EM algorithm:

[0119]

[0120] The statistical model obtains parameters After that, substitute X into the statistical model to obtain a probability matrix of the same size as X:

[0121]

[0122] Finally, use Through a weighted equation, the preliminary interpolation data X′ of the GCN model is constrained with reference to X to obtain the final interpolation data

[0123]

[0124] By constraining the preliminary interpolation result X′ in 4), the final interpolation result of Raw data is obtained. The process is shown in Algorithm 4:

[0125]

[0126] At this point, the entire interpolation process is completed.

[0127] The comparison results between this example and the existing method are as follows Figure 1-Figure 3 As shown in the figure, the interpolation data obtained by this method has a better clustering effect and higher interpolation accuracy than the interpolation results of other comparison methods, and can better restore the true expression pattern of the data.

Claims

1. A scRNA-seq data interpolation method based on graph convolutional neural networks and statistical model constraints, characterized in that: The steps include: 1) Data preprocessing: The scRNA-seq data is used as input, and the input scRNA-seq data is preprocessed for data filtering and quality control: first, only genes expressed in more than 1% of cells and cells expressed in more than 1% of genes are retained; second, if the number of retained genes is too large, the retained genes are sorted by standard deviation, and the top 1000 genes with the largest standard deviation are selected for research, thus obtaining the scRNA-seq data X for input into the model; 2) Graph information construction and pre-label acquisition: Based on GCN, preliminary interpolation is performed, so the graph information between cell samples, that is, the adjacency matrix A between cells, is required as input. The K nearest neighbor algorithm, that is, KNN, is used to construct the graph information. In order to avoid the noise in X and provide accurate information for the subsequent construction of the adjacency matrix, PCA dimensionality reduction is first performed on X: X d =PCA(X) (1) Then use the K nearest neighbor algorithm to find X d Construct the adjacency matrix: A=KNN(X d ) (2) The top-k parameter of KNN is set to 5, and then d Perform pre-clustering to obtain the pre-label of each sample point: y p =KMeans(X d ) (3) At this point, all the input data required for GCN training is prepared; 3) Preliminary interpolation based on GCN: In order to identify dropout events in scRNA-seq data and interpolate them, a GCN-based model was trained. First, the adjacency matrix A was processed to obtain a normalized Laplacian matrix: in I represents the identity matrix of the same size as A, and Finally, input X and the normalized Laplacian adjacency matrix into GCN and obtain preliminary interpolation results: Where W i represents the weight parameter of the i-th layer of GCN, σ represents the activation function of each layer, and in formula (5), we use Rather than The reason is that in the iteration of training GCN, the initial normalized Laplacian matrix is ​​not always used. Instead, as the iteration proceeds, the better data X′ output by the model is used to construct a better normalized Laplacian adjacency matrix in the order of formulas (1), (2), and (4). For subsequent iterative training, Because GCN is still not accurate enough in reconstructing and denoising scRNA-seq data, we added a design based on GCN that enables GCN to accurately identify dropout events and make reasonable interpolation: 3.1) In order to make the GCN model not make changes to the places with expression values ​​in X, a separate mean square error loss function is designed for the places with expression values ​​in X, and the places with expression values ​​in X are recorded as 1 to obtain a labeling matrix e of the same size as X: And use the loss function L1 to guide the training of the corresponding weights: L1=MSE(X·e,X′·e) (7) 3.2) Since the dropout event only occurs where the expression value is zero in X, in order to make the initial interpolation of the GCN model be performed where the expression value is 0, the strategy of step 3.1) is adopted to mark the places where the expression value is 0 in X with 1, and obtain a marking matrix of the same size as X And use the loss function L2 to guide the training of the corresponding weights: Among them, α is a tolerance factor specially designed for the loss function L2, and its range is [0,1). The larger α is, the larger the interpolation value is where the expression value is 0. σ is a Gaussian kernel function parameter, which can be calculated by the number of cells with expression values ​​0 in X and the initialized α. The following are the steps to calculate the parameter σ: Assuming that the size of the existing X to be interpolated is n*m, and there are p 0 values ​​in X, then the proportion of 0 values ​​in X is: γ=p / (n*m), then: min`error=(p*λ*a) 2 (10) max`error=(p*λ*b) 2 (11) in λ indicates the number of zero values ​​that need to be interpolated. Since scRNA-seq data is sparse, λ is set to a small value in order not to destroy the inherent laws of the data. In addition, a and b indicate the range of values ​​that need to be interpolated at zero, which can also be expressed as [a, b]. a and b are hyperparameters that can be set by yourself. Then, the X-specific σ value is calculated according to the following equation and algorithm flow: 3.3) The data processed by GCN will have the phenomenon of overfitting. In order to overcome this problem, a classifier is added to the bottleneck layer of the GCN in the shape of the autoencoder. The classifier can alleviate the overfitting phenomenon through the preset hyperparameter K, and by presetting a classification loss function L3, the final interpolation result and downstream analysis can obtain good performance. The loss function L3 is used to guide the training of the corresponding weights: L3=CrossEntropy(y p ,and) (14) where y p is the pre-label of each sample obtained in step 2), y is the multi-classification value obtained by the classifier, and the classifier is composed of a fully connected neural network, whose input Z is the output of the bottleneck layer of the autoencoder: Get y p The number of clusters K of KMeans is actually unknown, so a list K = [n-1, n, n+1] is set up. p When , the three cluster numbers in K are used respectively. Similarly, in order to p Correspondingly, the number of neurons in the output layer of the classifier is also changed to the number in K. This strategy means that the GCN model needs to be trained three times with different numbers of clusters. The final estimated data given by GCN is the average of these three estimates, where n is a user-defined hyperparameter. After the above steps 3.1), 3.2) and 3.3), the final loss function L of GCN is obtained: L=β*L1+(1-β)*(L2+L3) (16) β is a custom parameter; 4) Statistical model constraints based on negative binomial or zero-inflated negative binomial distribution: After the preliminary interpolation in step 3), in order to prevent GCN from incorrectly interpolating the true zero value, the negative binomial or zero-inflated negative binomial distribution statistical model is used to further constrain the interpolation results: Formula (17) is the probability mass equation of the negative binomial distribution, where X is the scRNA-seq data, also called the observed variable in formula (17), ф and μ are the parameters of the negative binomial distribution. On the basis of the NB distribution, the probability mass equation of the ZINB distribution can be obtained by adding the zero inflation factor: Through the negative binomial or zero-inflated negative binomial distribution statistical model, the probability of the expression value appearing in X under the corresponding distribution can be calculated. The specific estimation method used is the EM algorithm: The statistical model obtains parameters After that, substitute X into the statistical model to obtain a probability matrix of the same size as X: Finally, use Through a weighted equation, the preliminary interpolation data X′ of the GCN model is constrained with reference to X to obtain the final interpolation data At this point, the entire interpolation process is completed.

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