Adaptive topology recovery and connection weight estimation method for dynamic network with hidden nodes

Abstract

The application discloses a self-adaptive topology recovery and connection weight estimation method for a dynamic network containing hidden nodes, and belongs to the technical field of network system modeling. Firstly, only two convex optimization problems need to be solved at each time, which significantly improves the calculation efficiency; secondly, the system parameter estimation is decoupled from the hidden variable dimension estimation and topology estimation, avoiding the influence of the regularization term on the parameter estimation accuracy; finally, by introducing a hidden variable parameter estimation method based on matrix Cholesky decomposition, the orthogonal transformation error in traditional spectral decomposition is effectively overcome, thereby ensuring the consistency of parameter estimation.

Description

Technical Field

[0001] This invention relates to the field of networked system modeling technology, and designs an adaptive topology restoration and connection weight estimation method for dynamic networks with hidden nodes, realizing network topology reconstruction and parameter estimation. Background Technology

[0002] Simplified modeling of high-dimensional stationary random sequences can often be transformed into a sparse dynamic network reconstruction problem with latent variables. Different dimensions of the variables correspond to different network nodes. There are currently two mainstream ways to define the topology between nodes. One is the conditional dependency defined by the sparse structure of the inverse power spectral density matrix, which corresponds to an undirected network. This type of research mainly focuses on whether there is a statistical conditional correlation between variables. The other is the Granger causality defined by the necessity of one variable for reconstructing another variable, which corresponds to a directed network. This type of research mainly focuses on whether the current and past information of one variable helps to predict the future of another variable, which is a direct influence in the time domain.

[0003] Considering that the characterization of directed graphs naturally encompasses the characterization of undirected graphs in practice, we utilize Granger causality to define the topological relationships between nodes. Under this topological definition, many network parameter estimation algorithms exist based on the idea of ​​minimizing prediction errors when the topology is known. Further research has focused on dynamic network systems with low-rank noise processes, using multi-step sequential least squares and weighted null space fitting methods to estimate the topology and parameters of noise variables, achieving overall network parameter estimation when the topology of the remaining nodes is known. Considering that noise variables are also unobservable in reality, the noise variables in the above studies can essentially be regarded as hidden nodes in the network. Furthermore, for cases where latent variables can be determined by historical manifest variables, some literature proposes a kernel regularization method based on a sparse and low-rank framework to estimate network topology and parameters. However, the above methods cannot be directly applied when unknown latent variable sequences and unknown noise sequences coexist, and the latent variable sequence does not depend on the historical manifest variable sequence.

[0004] Currently, there is a lack of direct research on directed topology and parameter estimation of networks. The research by You et al. on autoregressive undirected graphical models with dynamic latent variables is somewhat relevant, and their study proposed the Sparse Plus Low Rank Identification (SPLRID) algorithm. Specifically, this algorithm will... Alternating iterative solutions to the norm regularization term and the convex optimization problem with the kernel norm regularization term achieve joint estimation of network parameters and topology. However, this algorithm is designed for undirected network topology and is not applicable to directed networks. Furthermore, its alternating iterative strategy consumes significant computation time when the network size is large. Therefore, we are interested in designing an efficient reconstruction method for undirected networks with dynamic latent variables. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide an adaptive topology restoration and connection weight estimation method for dynamic networks with hidden nodes, which can alleviate the computational burden of existing algorithms iteratively solving several optimization problems and avoid estimation biases introduced by regularization terms.

[0006] An adaptive topology recovery and connectivity weight estimation method for dynamic networks with hidden nodes is proposed. This method employs a three-stage identification framework to jointly estimate the parameters of observed variables and the topology, as well as the dimension of hidden variables and their parameters in relation to the topology.

[0007] The first step is to establish a recursive estimate of the dynamic parameters of the observed variables based on the sample correlation function of the system measurements; then, to reconstruct the topology between the observed nodes through an adaptive threshold truncation strategy; and finally, to construct a system with local... A convex optimization problem with regularization terms is solved to improve the estimation accuracy of dynamic parameters of observed variables; The second step involves using the parameter estimates obtained in the first step to construct a constrained convex optimization problem based on the power spectral density identity, estimating the dynamic parameters of the latent variables coupled in a product form; then, an adaptive eigenvalue truncation strategy is designed to estimate the dimension of the latent variables. The third step involves first performing eigenvalue decomposition on the dynamic parameters coupled in a product form, then truncating the eigenvalues ​​based on the latent variable dimension estimation, and subsequently using matrix Cholesky decomposition to decouple the truncated coupled estimates, thereby obtaining the estimates of the latent variable dynamic parameters. On this basis, an adaptive threshold truncation strategy is used to restore the topological structure from latent variables to observed variables, and the latent variable dynamic parameter estimates are further updated in conjunction with the existing parameter estimates. Through the above three stages, the parameters and topology of a dynamic network system containing dynamic hidden variables are reconstructed.

[0008] Preferably, in the first step, the recursive estimation of the dynamic parameters of the observed variables is achieved in the following way: Consider dynamic network systems with latent variables:

[0009] in, , , They are respectively The system outputs observation data, latent variables, and noise at various times. and These are the dimensions of the observed data and the latent variables, respectively. For the shift operator, i.e. , is used to represent the value of a variable at a past time; where, express The observation data output by the time system; Matrix polynomial ; Among them, the coefficients of the two matrix polynomials , and coefficient The parameters to be identified and the matrix The former The submatrix formed by the rows is a lower triangular matrix containing elements on the positive diagonal, and the integers are positive. and The system order is known. express 3D identity matrix; For measurable nodes in the observation data Defined as a set of elements consisting of 0 and 1. Topological connectivity matrix between observable variables , its first Line 1 Column position element Indicates from measurable nodes To the measurable node There is a connection, if Indicates from measurable nodes To the measurable node No connection; Define the process from hidden nodes to measurable nodes. 3D topological connectivity matrix , its first Line 1 Column position element Indicates from hidden nodes To the measurable node There is a connection, if Indicates from hidden nodes To the measurable node No connection; From the current moment Observational data At this historic moment Observational data Stack them into a long vector, and denote it as , where positive integers The system order is defined earlier; similarly, it will be from time... Observational data At this historic moment Observational data Stack them into a long vector, and denote it as From time Observational data At this historic moment Observational data Stack them into a long vector, and denote it as Based on the correlation functions of samples of various orders of measurable variables, calculate the dynamic parameter matrix of the observed variables. The following algorithm estimates the time... Running online, and These are used to calculate the parameter matrix. estimate intermediate variables, and Each variable corresponds to a time interval. The corresponding values, and Each variable corresponds to a time interval. The corresponding values; setting the initial time. , ;

[0010] Preferably, in the first step, reconstructing the topology between observation nodes using an adaptive threshold truncation strategy specifically includes: For any index Select an adaptive threshold The calculation is performed according to the following truncation strategy:

[0011] in, These are hyperparameters given in advance; Represents the parameter matrix exist Elements at the location; The elements obtained above Composition Each position in the middle and Correspondingly, we can obtain the matrix polynomial. Parameter estimation at each order Defined by the transpose of each row-divided submatrix of the matrix, i.e. ,in Representation matrix The arrive The submatrix formed by rows, similarly, Representation matrix The arrive The submatrix formed by rows; then the topological estimate between observable variables is calculated. Its elements are represented as follows: .

[0012] Preferably, in the first step, the estimated topology is used to construct a local topology. The convex optimization problem of regularization term is solved to improve the estimation accuracy of dynamic parameters of observed variables, specifically including: Selecting regularization coefficients Using existing estimates Given a sparse structure, we establish the following convex optimization problem:

[0013] in, Representation matrix The complement of the support set, i.e., the set of zero element positions; the convex optimization problem is described as: [The following text appears to be a list of variables and their complements, and is not directly related to the previous sentence.] While ensuring that the positions of non-zero elements remain consistent with existing estimates of non-zero element positions, the mean square loss term is minimized, i.e., the variable that best fits the known algebraic relationship is selected. This allows for more accurate parameter estimation. .

[0014] Preferably, in the second step, estimating the dynamic parameters of the latent variables coupled in a product form specifically includes: Estimating the dynamic parameter matrix of product-form coupling ,in, Based on the dynamic network system equations, the power spectral density matrix identity is derived, and the algebraic relationships between the parameters of each order are established. Then, the following estimation scheme is designed: From the current moment Observational data At this historic moment Observational data Stack them into a long vector, and denote it as From time Observational data At this historic moment Observational data Stack them into a long vector, and denote it as For any index ,matrix For the two intermediate variables of the algorithm running online at time... The value of , For this intermediate variable at time... The value of ; Select the initial time Using existing estimates calculate:

[0015]

[0016] Furthermore, using the CVX toolbox in MATLAB to solve the following constrained convex optimization problem, we obtain the following results. Estimate :

[0017]

[0018] in, Represents a block matrix The Diagonal submatrices; Represents a block matrix The Line 1 Column submatrix.

[0019] Preferably, in the second step, estimating the dimension of the latent variables specifically includes: calculate Eigenvalue decomposition:

[0020] in, Indicates sorted in descending order eigenvalues, It is a diagonal matrix composed of eigenvalues; for A matrix consisting of eigenvectors whose eigenvalues ​​are arranged in the same order; Set adaptive cutoff threshold For any , will estimate The eigenvalues ​​shrink in the following form:

[0021] in, express The Middle One element; The corresponding contracted matrix estimate is obtained:

[0022] Among them, hyperparameters ; Finally, calculate the estimated value of the hidden variable dimension: .

[0023] Preferably, in the third step, the estimation of the latent variable dynamic parameters is obtained, specifically including: right Before proceeding The eigenvalues ​​were truncated to obtain:

[0024] in, For the reason The former A diagonal matrix composed of eigenvalues; for forward A matrix composed of columns; extract The former row, before Submatrix, denoted as ; extract After row, before Submatrix, denoted as ; Calculate the positive definite matrix The Cholesky decomposition is performed, and the resulting matrix factorization factors are denoted as:

[0025] This is also a parameter The former The corresponding estimate is then calculated; based on this, the parameters are further calculated. back Line estimation:

[0026] Together, they constitute the latent variable dynamic parameters. Overall estimate:

[0027] Preferably, in the third step, an adaptive threshold truncation strategy is used to restore the topological structure from latent variables to observed variables, specifically including: For any Select the regularization coefficient , Update the latent variable parameter estimates for pre-given hyperparameters. ,as follows:

[0028] in, express Middle position Elements at the location; Parameter estimation of each order Defined as matrices respectively Each column of the submatrix ; Minimize Cross-validation confirms the result. Then, the topological estimate from the latent variables to the observed variables is calculated. Its elements are represented as follows: .

[0029] Preferably, in the third step, the latent variable dynamic parameter estimates are further updated, specifically including: Using existing estimates Sparse structure, updating latent variable parameter estimates ,as follows: .

[0030] Better, comprehensive estimation of measurable variable parameters With topology estimation Latent variable dimensionality estimation And parameter estimation from latent variables to measurable variables. With topology estimation This completes the full reconstruction of the network system.

[0031] The present invention has the following beneficial effects: This invention proposes a novel identification framework for autoregressive network systems modeled with dynamic latent variables. This framework is based on the correlation function of observable samples and the statistical characteristics of latent variables and noise. The framework has the following characteristics: First, it requires solving only two convex optimization problems at each time step, significantly improving computational efficiency. Second, it decouples system parameter estimation from latent variable dimension estimation and topology estimation, avoiding the impact of regularization terms on parameter estimation accuracy. Finally, by introducing a latent variable parameter estimation method based on matrix Cholesky decomposition, it effectively overcomes the orthogonal transformation error in traditional spectral decomposition, thus ensuring the consistency of parameter estimation. Attached Figure Description

[0032] Figure 1 A schematic diagram of a directed network containing 2 hidden nodes and 3 observable nodes; Figure 2 This indicates the parameter estimation error of the algorithm in this invention (directed network). Figure 3 This indicates the topology recovery accuracy of the algorithm in this invention (directed network). Figure 4 This represents the accuracy of the hidden variable dimension recovery algorithm in this invention (directed network). Figure 5 This invention represents the algorithm for network structure estimation (directed network). Figure 6A schematic diagram of an undirected network containing 1 hidden node and 10 observable nodes; Figure 7 This indicates the parameter estimation error of the algorithm in this invention (undirected network). Figure 8 This indicates the topology recovery accuracy of the algorithm in this invention (undirected network). Figure 9 This indicates the accuracy of the hidden variable dimension recovery algorithm in this invention (undirected network). Figure 10 This section compares the performance of the algorithm presented in this invention with that of the SPLRID algorithm. Detailed Implementation

[0033] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0034] This invention proposes an adaptive topology recovery and connectivity weight estimation method for dynamic networks with hidden nodes. It employs a three-stage identification framework to jointly estimate the parameters of observed variables and the topology, as well as the dimension of hidden variables and their parameters in relation to the topology. In the first stage, based on the sample correlation function of system measurements, a recursive estimate of the dynamic parameters of the observed variables is established; the topological structure between the observed nodes is reconstructed through an adaptive threshold truncation strategy; and then, using the estimated topological structure, a system with local... The convex optimization problem of regularization term is solved to improve the estimation accuracy of dynamic parameters of observed variables.

[0035] In the second stage, using the parameter estimates obtained in the first stage, a constrained convex optimization problem is constructed based on the power spectral density identity to estimate the dynamic parameters of the latent variables coupled in a product form; then, an adaptive eigenvalue truncation strategy is designed to estimate the dimension of the latent variables.

[0036] In the third stage, the coupled parameter estimates are first decomposed into eigenvalues, and then the eigenvalues ​​are truncated based on the latent variable dimension estimates. Subsequently, the matrix Cholesky decomposition technique is used to decouple the truncated coupled estimates, thereby obtaining the estimates of the latent variable dynamic parameters. On this basis, an adaptive threshold truncation strategy is used to restore the topological structure from latent variables to observed variables, and the dynamic parameter estimates of the latent variable part are further updated by combining the existing parameter support set estimates.

[0037] Through the above three stages, an adaptive topology restoration and connectivity weight estimation method for dynamic networks with hidden nodes is realized. The specific scheme is as follows: Consider dynamic network systems with latent variables:

[0038] in, , , They are respectively The system outputs observation data, latent variables, and noise at various times. and The dimensions of the observed data and the latent variables are respectively... . and They are independent Gaussian white noise sequences, and their power spectral density matrices are identity matrices. For the shift operator, i.e. , is used to represent the value of a variable at a past time; where, express The observation data output by the time system; Matrix polynomial ; Among them, the coefficients of the two matrix polynomials , and coefficient The parameters to be identified and the matrix The former The submatrix formed by the rows is a lower triangular matrix containing elements on the positive diagonal, and the integers are positive. and The system order is known. express 3D identity matrix.

[0039] Note the measurable variables for dimensional vector, its Each dimension corresponds to a different aspect of the network system. observable nodes Latent variables for dimensional vector, its Each dimension corresponds to a different aspect of the network system. Hidden nodes The topology between network nodes is defined according to Granger causality: if a node contributes to the modeling accuracy of another node, there is a connection from the former to the latter. Specifically, for measurable nodes in the observation data... Defined as a set of elements consisting of 0 and 1. Topological connectivity matrix between observable variables , its first Line 1 The element at the column position is represented as:

[0040] in, According to the above definition, if Then from the measurable nodes To the measurable node There is a connection, if Then from the measurable nodes To the measurable node No connection.

[0041] Similarly, it is possible to define from hidden nodes To the measurable node of 3D topological connectivity matrix , its first Line 1 The element at the column position is represented as:

[0042] For any index This definition indicates that if Then from the hidden node To the measurable node There is a connection, if Then from the hidden node To the measurable node No connection.

[0043] The problem addressed by this invention for the aforementioned system can be described as: utilizing only system measurement data Recovering the dimension of hidden variables , parameter matrix , and topology connection matrix and .

[0044] The network parameter and topology joint identification algorithm proposed in this invention consists of the following steps: Step 1: Estimate the parameter matrix .

[0045] From the current moment Observational data At this historic moment Observational data Stack them into a long vector, and denote it as , where positive integers The system order is defined earlier; similarly, it will be from time... Observational data At this historic moment Observational data Stack them into a long vector, and denote it as From time Observational data At this historic moment Observational data Stack them into a long vector, and denote it as Based on the correlation functions of samples of various orders of measurable variables, calculate the dynamic parameter matrix of the observed variables. The following algorithm estimates the time... Running online, and These are used to calculate the parameter matrix. estimate intermediate variables, and Each variable corresponds to a time interval. The corresponding values, and Each variable corresponds to a time interval. The corresponding values; setting the initial time. , ,but:

[0046] Step 2: Estimate the topology connectivity matrix .

[0047] For any index Select an adaptive threshold The calculation is performed according to the following truncation strategy:

[0048] in, The hyperparameters are given in advance, and the hyperparameters are... Minimize Cross-validation confirmed the results. Represents the parameter matrix exist Elements at the location; The elements obtained above Composition Each position in the middle and Correspondingly, we obtain the matrix polynomial. Parameter estimation at each order Each of these is defined by the transpose of the row-level submatrices of the matrix, i.e. ,in Representation matrix The arrive Submatrix formed by rows, Representation matrix The arrive The submatrix formed by rows; then the topological estimate between observable variables is calculated. Its elements are represented as follows:

[0049] Step 3: Improve parameters The accuracy of the estimation.

[0050] Selecting regularization coefficients Using existing estimates Given a sparse structure, we establish the following convex optimization problem:

[0051] in, Representation matrix The complement of the support set, i.e., the set of zero element positions; the convex optimization problem is described as: [The following text appears to be a list of variables and their complements, and is not directly related to the previous sentence.] While ensuring that the positions of non-zero elements remain consistent with existing estimates of non-zero element positions, the mean square loss term is minimized, i.e., the variable that best fits the known algebraic relationship is selected. This allows for more accurate parameter estimation. The above optimization problem can be solved using the CVX toolbox in MATLAB.

[0052] Step 4: Estimate the product-form coupling parameter matrix ,in, Based on the dynamic network system equations, the power spectral density matrix identity is derived, and the algebraic relationships between the parameters of each order are established. Then, the following estimation scheme is designed: From the current moment Observational data At this historic moment Observational data Stack them into a long vector, and denote it as From time Observational data At this historic moment Observational data Stack them into a long vector, and denote it as For any index ,matrix intermediate variables for online algorithm execution and At any moment The value of , intermediate variables At any moment The value of ; Select the initial time Using existing estimates calculate:

[0053]

[0054] Furthermore, the coupling parameters were obtained by solving the following constrained convex optimization problem using the CVX toolbox in MATLAB. Estimate :

[0055]

[0056] in, Represents a block matrix The Diagonal submatrices; Represents a block matrix The Line 1 Column submatrix.

[0057] Step 5: Estimate the dimension of latent variables .

[0058] Calculate a symmetric positive semidefinite matrix eigenvalue decomposition

[0059] in, Indicates sorted in descending order eigenvalues, It is a diagonal matrix composed of eigenvalues; for A matrix consisting of eigenvectors whose eigenvalues ​​are arranged in the same order; Set adaptive cutoff threshold , These are hyperparameters given in advance. For any , will estimate The eigenvalues ​​shrink in the following form:

[0060] in, express The Middle One element; The corresponding contracted matrix estimate is obtained:

[0061] Among them, hyperparameters It can be minimized Cross-validation selection, and .

[0062] Finally, calculate the estimated value of the hidden variable dimension: .

[0063] Step 6: Estimate the latent variable parameter matrix .

[0064] right Before proceeding The eigenvalues ​​were truncated to obtain:

[0065] in, For the reason The former A diagonal matrix composed of eigenvalues; for forward A matrix composed of columns; extract The former row, before Submatrix, denoted as ; extract After row, before Submatrix, denoted as ; Calculate the positive definite matrix The Cholesky decomposition is performed, and the resulting matrix factorization factors are denoted as:

[0066] This is also a parameter The former The corresponding estimate is then calculated; based on this, the parameters are further calculated. back Line estimation:

[0067] Together, they constitute the latent variable dynamic parameters. Overall estimate:

[0068] Step 7: Estimate the topology connectivity matrix .

[0069] For any Select the regularization coefficient , Update the latent variable parameter estimates for pre-given hyperparameters. ,as follows:

[0070] in, express Middle position The element value at that position; Parameter estimation of each order Defined as matrices respectively Each column of the submatrix ; Minimize Cross-validation confirms the result. Then, the topological estimate from the latent variables to the observed variables is calculated. Its elements are represented as follows: .

[0071] Step 8: Increase the dynamic parameters of hidden variables The accuracy of the estimation.

[0072] Using existing estimates Sparse structure, updating latent variable parameter estimates ,as follows: .

[0073] Step 9: Network system reconstruction.

[0074] Comprehensive Measurable Variable Parameter Estimation With topology estimation Latent variable dimensionality estimation And parameter estimation from latent variables to measurable variables. With topology estimation This completes the full reconstruction of the network system.

[0075] For the attached figure Figure 1 The directed network system shown is used to verify the algorithm proposed in this invention through 100 Monte Carlo experiments. Its parameter estimation error, topology recovery accuracy, and latent variable dimension recovery accuracy are shown in the attached figures. Figure 2-4 The effectiveness of the algorithm is illustrated in the diagram. Additionally, the attached diagram... Figure 5 The paper also presents a comparison between the structure estimation and the actual structure, demonstrating the accuracy of the algorithm in recovering the entire network structure (including global topology and the dimension of latent variables).

[0076] For the attached figure Figure 6 The undirected network system shown is used to verify the algorithm proposed in this invention through 100 Monte Carlo experiments. The parameter estimation error, topology recovery accuracy, and latent variable dimension recovery accuracy are shown in the attached figures. Figure 7-9 The results are given in [the document]. Furthermore, we compared the algorithm of this invention with the existing SPLRID algorithm; both can correctly recover the topology of the network system (e.g., [the algorithm is described in the document]). Figure 10 As shown in Table 1), the algorithm of this invention has significant improvements in parameter estimation accuracy and computational efficiency.

[0077] Table 1

[0078] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. An adaptive topology restoration and connectivity weight estimation method for dynamic networks with hidden nodes, characterized in that, A three-stage identification framework is used to jointly estimate the parameters and topological structure of observed variables, the dimension of latent variables and their parameters and topology: The first step involves establishing a recursive estimate of the dynamic parameters of the observed variables based on the sample correlation function of the system measurements; reconstructing the topology between the observed nodes using an adaptive threshold truncation strategy; and then utilizing the estimated topology information to construct a system with local... A convex optimization problem with regularization terms is solved to improve the estimation accuracy of dynamic parameters of observed variables; The second step involves using the parameter estimates obtained in the first step to construct a constrained convex optimization problem based on the power spectral density identity, estimating the dynamic parameters of the latent variables coupled in a product form; then, an adaptive eigenvalue truncation strategy is designed to estimate the dimension of the latent variables. The third step involves first performing eigenvalue decomposition on the dynamic parameters coupled in a product form, then truncating the eigenvalues ​​based on the latent variable dimension estimation, and subsequently using matrix Cholesky decomposition to decouple the truncated coupled estimates, thereby obtaining the estimates of the latent variable dynamic parameters. On this basis, an adaptive threshold truncation strategy is used to restore the topological structure from latent variables to observed variables, and the latent variable dynamic parameter estimates are further updated in conjunction with the existing parameter estimates. Through the above three stages, the parameters and topology of a dynamic network system containing dynamic hidden variables are reconstructed.

2. The adaptive topology restoration and connectivity weight estimation method for dynamic networks with hidden nodes as described in claim 1, characterized in that, In the first step, the recursive estimation of the dynamic parameters of the observed variables is achieved in the following way: Consider dynamic network systems with hidden variables: in, , , They are respectively The system outputs observation data, latent variables, and noise at various times. and These are the dimensions of the observed data and the latent variables, respectively. For the shift operator, i.e. , is used to represent the value of a variable at a past time; where, express The observation data output by the time system; Matrix polynomial ; Among them, the coefficients of the two matrix polynomials , and coefficient The parameters to be identified and the matrix The former The submatrix formed by the rows is a lower triangular matrix containing elements on the positive diagonal, and the integers are positive. and The system order is known. express 3D identity matrix; For measurable nodes in the observation data Defined as a string consisting of 0 and 1 elements. Topological connectivity matrix between dimensional observable variables , its first Line number Column position element Indicates from measurable nodes To the measurable node There is a connection, if Indicates from measurable nodes To the measurable node No connection; Define the transition from hidden nodes to measurable nodes. 3D topological connection matrix , its first Line number Column position element Indicates from hidden nodes To the measurable node There is a connection, if Indicates from hidden nodes To the measurable node No connection; From the current moment Observational data At this historic moment Observational data Stack them into a long vector, and denote it as , where positive integers The system order is defined earlier; similarly, it will be from time... Observational data At this historic moment Observational data Stack them into a long vector, and denote it as From time Observational data At this historic moment Observational data Stack them into a long vector, and denote it as Based on the correlation functions of samples of various orders of measurable variables, calculate the dynamic parameter matrix of the observed variables. The following algorithm estimates the time... Running online, and These are used to calculate the parameter matrix. estimate intermediate variables, and Each variable corresponds to a time interval. The corresponding values, and Each variable corresponds to time. The corresponding values; setting the initial time. , , 。 3. The adaptive topology restoration and connectivity weight estimation method for dynamic networks with hidden nodes according to claim 2, characterized in that, In the first step, the topology between observation nodes is reconstructed using an adaptive threshold truncation strategy, specifically including: For any index Select an adaptive threshold The calculation is performed according to the following truncation strategy: in, These are hyperparameters given in advance; Represents the parameter matrix exist Elements at the location; The elements obtained above Composition Each position in the middle and Correspondingly, we can obtain the matrix polynomial. Parameter estimation at each order Defined by the transpose of each row-divided submatrix of the matrix, i.e. ,in Representation matrix The arrive The submatrix formed by rows, similarly, Representation matrix The arrive The submatrix formed by rows; then the topological estimate between observable variables is calculated. Its elements are represented as follows: 。 4. The adaptive topology restoration and connection weight estimation method for dynamic networks with hidden nodes according to claim 3, characterized in that, In the first step, the estimated topology is used to construct a local topology. The convex optimization problem of regularization term is solved to improve the estimation accuracy of dynamic parameters of observed variables, specifically including: Selecting regularization coefficients Using existing estimates Given a sparse structure, we establish the following convex optimization problem: in, Representation matrix The complement of the support set, i.e., the set of zero element positions; the convex optimization problem is described as: [The following text appears to be a list of variables and their complements, and is not directly related to the previous sentence.] While ensuring that the positions of non-zero elements remain consistent with existing estimates of non-zero element positions, the mean square loss term is minimized, i.e., the variable that best fits the known algebraic relationship is selected. This allows for more accurate parameter estimation. .

5. The adaptive topology restoration and connection weight estimation method for dynamic networks with hidden nodes according to claim 4, characterized in that, In the second step, the dynamic parameters of the latent variables, coupled in a product form, are estimated, specifically including: Estimating the dynamic parameter matrix of product-form coupling ,in, Based on the dynamic network system equations, the power spectral density matrix identity is derived, and the algebraic relationships between the parameters of each order are established. Then, the following estimation scheme is designed: From the current moment Observational data At this historic moment Observational data Stack them into a long vector, and denote it as From time Observational data At this historic moment Observational data Stack them into a long vector, and denote it as For any index ,matrix For the two intermediate variables of the algorithm running online at time... The value of , For this intermediate variable at time... The value of ; Select the initial time Using existing estimates calculate: Furthermore, using the CVX toolbox in MATLAB to solve the following constrained convex optimization problem, we obtain the following results. Estimate : ； in, Represents a block matrix The Diagonal submatrices; Represents a block matrix The Line number Column submatrix.

6. The adaptive topology restoration and connection weight estimation method for dynamic networks with hidden nodes according to claim 5, characterized in that, In the second step, estimating the dimension of the latent variables specifically includes: calculate Eigenvalue decomposition: in, Indicates sorted in descending order eigenvalues, It is a diagonal matrix composed of eigenvalues; for A matrix consisting of eigenvectors whose eigenvalues ​​are arranged in the same order; Set adaptive cutoff threshold For any , will estimate The eigenvalues ​​shrink in the following form: in, express The Middle One element; The corresponding contracted matrix estimate is obtained: Among them, hyperparameters ; Finally, calculate the estimated value of the dimension of the latent variables: .

7. The adaptive topology restoration and connection weight estimation method for dynamic networks with hidden nodes according to claim 6, characterized in that, The third step involves obtaining estimates of the latent variable dynamic parameters, specifically including: right Before proceeding The eigenvalues ​​are truncated to obtain: in, For the reason The former A diagonal matrix composed of eigenvalues; for forward A matrix composed of columns; extract The former row, before Submatrix, denoted as ; extract After row, before Submatrix, denoted as ; Calculate the positive definite matrix The Cholesky decomposition is performed, and the resulting matrix factorization factors are denoted as: This is also a parameter The former The corresponding estimate is then calculated; based on this, the parameters are further calculated. back Line estimation: Together, they constitute the latent variable dynamic parameters. Overall estimate: 。 8. The adaptive topology restoration and connectivity weight estimation method for dynamic networks with hidden nodes according to claim 7, characterized in that, In the third step, an adaptive threshold truncation strategy is used to restore the topological structure from latent variables to observed variables, specifically including: For any Select the regularization coefficient , Update the latent variable parameter estimates for pre-given hyperparameters. ,as follows: in, express Middle position The element value at the specified location; Each position in the middle and Correspondingly, we can obtain Parameter estimation at each order Then, the topological connectivity matrix estimate from the latent variables to the observed variables is calculated. Its elements are represented as follows: 。 9. The adaptive topology restoration and connectivity weight estimation method for dynamic networks with hidden nodes according to claim 8, characterized in that, The third step further updates the latent variable dynamic parameter estimates, specifically including: Using existing estimates Sparse structure, updating latent variable parameter estimates ,as follows: 。 10. The adaptive topology restoration and connectivity weight estimation method for dynamic networks with hidden nodes according to claim 9, characterized in that, Comprehensive Measurable Variable Parameter Estimation With topology estimation Estimation of the number of latent variables And parameter estimation from latent variables to measurable variables. With topology estimation This completes the full reconstruction of the network system.

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