A causal structure identification method for non-stationary under-sampling time series data

Abstract

The application provides a causal structure identification method for non-stationary under-sampling time series data, and relates to the technical field of causal discovery. Non-stationary under-sampling time series data is acquired and standardized, and a time series causal correlation model is constructed; for the current observation time step, the key hidden variables of the unobserved period are identified, and the time-lag causal structure is restored. The contemporaneous instantaneous causal structure is inferred, and the time series causal transmission structure pointing to the next unobserved period is constructed. The path separation set is constructed, and the data of the next observation time step is conditioned. The next observation time step is taken as the current observation time step, the conditioned data is taken as the current data, the causal structure inference is performed, and the process is repeated until all observation time steps are traversed. The time-lag causal structure, the contemporaneous instantaneous causal structure and the time series causal transmission structure of each observation time step are spliced to obtain a global causal structure graph. The application improves the accuracy of causal structure identification in the non-stationary under-sampling time series scene with hidden confusion.

Description

Technical Field

[0001] This invention belongs to the technical field of causal discovery, and more specifically, relates to a method for identifying causal structures in non-stationary undersampled time series data. Background Technology

[0002] Chronic diseases are typically characterized by long duration, slow progression, and numerous and complexly interacting influencing factors. Taking chronic kidney disease (CKD) as an example, patients often require extensive longitudinal follow-up. The relationships between renal function, blood pressure, blood glucose, inflammatory markers, and other physiological indicators may constantly change at different stages of the disease. For clinicians, simply knowing which indicators are abnormal at a particular point in time is usually insufficient; more importantly, it is crucial to identify the evolutionary relationships between these indicators throughout the entire disease course, thereby determining which factors warrant priority intervention. With the accumulation of electronic medical records, follow-up monitoring, and multi-timepoint clinical data, reconstructing key pathways in the disease evolution process from longitudinal observation has become a critical issue in the refined management and scientific intervention decision-making of chronic diseases.

[0003] Most existing medical data analysis methods focus on correlation analysis, risk prediction, or static structural modeling. In chronic kidney disease progression research, survival models such as Cox regression, recurrent neural networks (RNNs), and long short-term memory networks (LSTMs) are mostly used. These methods rely on time-series data of physiological indicators to achieve risk stratification of disease progression and prediction of end-stage renal disease, demonstrating excellent predictive accuracy. However, they can only characterize the statistical association between variables and clinical outcomes and cannot distinguish the driving and driven relationships between indicators, making it difficult to support the explanation of etiological mechanisms and the determination of intervention priorities.

[0004] To address the limitations of simple correlation analysis and predictive modeling, causal discovery methods are introduced into the analysis of complex time series data. Causal characterization methods based on time series prediction infer the directional relationship of time lags by examining whether past values ​​of a time series improve the prediction of another series. Dynamic Bayesian networks generalize Hidden Markov Models to sequential probabilistic graphical models, enabling the representation and inference of sequential data. Graph-based causal discovery has developed into several technical approaches, including constraint-based methods, scoring methods, and functional causal models, and has been applied in fields such as biomedicine.

[0005] Another type of approach enhances identifiability by introducing stronger distributional assumptions. Methods utilizing the non-Gaussian nature of noise can estimate the complete causal structure without pre-given causal order, but typically require the data generation process to meet strict assumptions of linearity, non-cyclicity, and lack of implicit confusion, which no longer holds when strong implicit confusion exists. For non-stationary data, methods introducing substitute variables to characterize distributional drift information can detect changes in local mechanisms and identify their direction. For strongly autocorrelated time series data, methods combining conditional independence tests with time series plot learning can handle lag and contemporaneous relationships, further considering potential confounding and strong autocorrelation.

[0006] However, directly applying the above methods to longitudinal follow-up of chronic diseases still faces several challenges. First, the mechanisms of action of chronic diseases at different stages are often not fixed, exhibiting significant non-stationary characteristics; that is, the causal relationships between the same set of indicators may change as the disease progresses. Second, clinical follow-up is usually limited by the duration of medical visits, testing frequency, and actual conditions, resulting in sparse sampling time points and significant undersampling issues, leading to unobserved processes between adjacent real-world conditions. Disease progression is often influenced by unmeasured physiological states, lifestyle, treatment adherence, and other potential factors, easily creating strong hidden confounding. Furthermore, clinically available and stable physiological indicators are often limited to a small number of low-dimensional physiological indicators. With limited samples and observational constraints, existing methods are more prone to problems such as inaccurate structure identification, unstable orientation judgment, or uninterpretable results.

[0007] In summary, existing causal structure identification methods are insufficient for analyzing the entire course of chronic diseases, which are characterized by non-stationary multi-stage disease progression, undersampling in longitudinal follow-up, strong concealment and confusion, and low-dimensional clinical indicators. This makes it difficult to recover the global causal structure of the entire course of the disease from local relationships, resulting in low accuracy of the identified causal structure and failing to provide technical solutions to support scientific intervention decisions. Summary of the Invention

[0008] To address the issue that existing causal structure identification methods have low accuracy when processing non-stationary undersampled time series data, this invention proposes a causal structure identification method for non-stationary undersampled time series data, which improves the accuracy of causal structure identification.

[0009] To achieve the above-mentioned technical effects, the technical solution of the present invention is as follows: S1: Obtain non-stationary undersampled time series data containing several observed variables, and standardize it to obtain standardized non-stationary undersampled time series data; construct a time series causal relationship model based on the standardized non-stationary undersampled time series data. S2: For the current observation time step, based on the current standardized non-stationary undersampled time series data, identify the key latent variables in the unobserved periods of the time series causal relationship model; based on the key latent variables, restore the time-delay causal structure of each observed variable in the current observation time step; S3: Based on the time-delay causal structure, infer the synchronous instantaneous causal structure among the observed variables at the current observation time step; based on the synchronous instantaneous causal structure, construct the temporal causal transmission structure of each observed variable at the current observation time step pointing to the next unobserved period. S4: Based on the synchronous instantaneous causal structure and the temporal causal transmission structure, construct the path separation set; use the path separation set to conditionally process the standardized non-stationary undersampled time series data of the next observation time step; S5: Using the next observation time step as the current observation time step, and using the standardized non-stationary undersampled time series data after conditional processing as the current standardized non-stationary undersampled time series data, repeat steps S2 to S4 until all observation time steps are processed. Then, splice the time-delay causal structure, the synchronous instantaneous causal structure, and the time-series causal propagation structure of each observation time step along the time sequence to obtain the global causal structure graph.

[0010] Furthermore, the process of constructing a time-series causal association model based on standardized non-stationary undersampled time-series data is as follows: Step 1: Set the subsampling rate of the non-stationary undersampled time series data to be... Set the observation time step set as ,in, Represents the set of observation time steps. This represents the R-th observation time step. Represents a fine-grained, full-time index set; Step 2: Set each observation time step to include Then the nth observed variable, The expression for the observation vector at each observation time step is:

[0011] In the formula, Indicates the first The observation vector for each observation time step. Indicates the first One observed variable; Step 3: Take the unobserved period between any two adjacent observation time steps in the standardized non-stationary undersampled time series data as the intermediate boundary period, and set the current observation time step... The output boundary is denoted as The next observation time step The input boundary is denoted as The expression for defining the boundary equivalence relation is:

[0012] In the formula, Indicates the next observation time step The input boundary, Indicates the current observation time step The output boundary; Step 4: Based on the boundary equivalence relation, construct a time-series causal association model, expressed as:

[0013]

[0014] In the formula, Indicates entering the current observation time step Unobserved input boundary, Indicates the current observation time step The output boundary points to the next unobserved time period. Indicates unobserved input boundary Observation vector at the current observation time step The time-delay causal structure, This represents the synchronous instantaneous causal structure within the current observation time step. This represents the temporal causal transit structure where each observed variable at the current observation time step points to the next unobserved time period. Represents the exogenous noise term. This represents the exogenous noise term.

[0015] Furthermore, based on the time-series causal association model and the current standardized non-stationary undersampled time-series data, the process of identifying key latent variables in the unobserved periods of the time-series causal association model is as follows: Step 1: Based on the current standardized non-stationary undersampled time series data, calculate the mixing coefficients of each observed variable at the current observation time step. The expression is:

[0016]

[0017] In the formula, Indicates higher-order cumulants. This represents the i-th observed variable at the current observation time step. This represents the j-th observed variable at the current observation time step. Represents observed variables The estimated mixing coefficients between the shared single key latent variable and the mixed coefficients. Represents observed variables Estimated mixing coefficients between the variable and a shared single key latent variable; Step 2: Combine any three observed variables in the current observation time into a triplet. Based on the mixing coefficient, determine each triplet in the current observation time step using the single latent component criterion. Combine the observed variables corresponding to the triplets that satisfy the single latent component criterion as the observed variable group. Each observed variable in the observed variable group shares the same key latent variable in the unobserved period. The observed variable set is expressed as follows:

[0018] In the formula, Represents the group of observed variables. Indicates the current observation time step The a-th observed variable, Indicates the current observation time step The One observed variable, Indicates the current observation time step The c-th observed variable; Step 3: Using each observed variable in the observed variable group as a proxy variable, perform regression on the other two observed variables in the observed variable group respectively to form a residual triplet, the expression of which is:

[0019] In the formula, Represents the residual triple. Represents a proxy variable. Represents proxy variables For observed variables The regression residuals after regression. Represents proxy variables For observed variables The regression residuals after regression; Step 4: If the residual triples satisfy the single latent component criterion, it indicates that the proxy variable... The source of influence is only the key latent variable; determine the proxy variable. This represents the pure child node of the key latent variable corresponding to the triple; Step 5: Identify the key hidden variables corresponding to nodes determined to be pure child nodes. and pure child nodes By using the same index for association, the key latent variables are uniquely determined in the observation variable space through the index of the pure child nodes, thus completing the identification of key latent variables in the unobserved period in the time-series causal association model.

[0020] Furthermore, the single latent component criterion is as follows: For the current observation time step Let i represent the observed variable. index, Indicates the removal of observed variables The index set of the remaining observed variables, and ; The core determination formula of the single hidden component criterion is:

[0021] In the formula, Indicates statistical independence. This represents a non-all-zero weight vector. This represents the coefficient vector obtained based on the mixed coefficient estimation. This represents all observed variables corresponding to index set J at the current observation time step. This represents the i-th observed variable in the current observation time step; When the core determination formula of the single latent component criterion is met, the corresponding variable group is determined to meet the single latent component criterion. The current variable group only shares a single key latent variable, and each observed variable in the variable group shares the same key latent variable in the unobserved period.

[0022] Furthermore, based on the key latent variables, the process of recovering the time-delay causal structure of each observed variable at the current observation time step is as follows: Step 1: For all observation variable groups that satisfy the single latent component criterion at the current observation time step, detect the pure child nodes corresponding to each observation variable group, associate the corresponding key latent variables through the pure child nodes, and establish a key latent variable group for each newly identified key latent variable. The key latent variable group contains the pure child nodes and coefficient vectors corresponding to the key latent variable. Step 2: Iteratively determine the remaining observation variable group. If the current observation variable group and the existing key latent variable group have at least one common observation variable, and the mixing coefficient of the common observation variable is consistent with the coefficient vector in the key latent variable group, then determine that the current observation variable group and the key latent variable group correspond to the same key latent variable. Add other observation variables in the current observation variable group that are not yet included in the key latent variable group to the key latent variable group, and recalculate the coefficient vector corresponding to the key latent variable group. Step 3: Based on the key latent variable groupings and coefficient vectors, determine the time-delay causal edges of each key latent variable pointing to each observed variable within the key latent variable grouping. For observed variables not covered by any key latent variable groupings, only the self-causal diagonal terms of the observed variable are retained to obtain the time-delay causal structure of the current observation time step.

[0023] Furthermore, based on the aforementioned time-delay causal structure, the process of inferring the synchronous instantaneous causal structure among the observed variables at the current observation time step is as follows: Step 1: Based on the time-delay causal structure, extract the set of pure child nodes and the set of non-pure child nodes in the current observation time step, and set all synchronous instantaneous causal edges corresponding to each pure child node in the set of pure child nodes to zero; Step 2: Form an unordered variable pair from any two non-pure child nodes in the set of non-pure child nodes. ,in, This represents the i-th observed variable in the current observation time step. Let represent the j-th observed variable in the current observation time step; determine whether each unordered variable pair shares a key latent variable. If they share a key latent variable, perform one-sided reduction on the unordered variable pair that shares the key latent variable and construct the adjusted observed variable pair as the variable pair to be tested; if they do not share a key latent variable, then the unordered variable pair is used as the variable pair to be tested. The process of constructing the adjusted pairs of observed variables is as follows: Select As a one-sided reduction object, the adjusted observed variable is constructed as follows:

[0024] In the formula, Indicates the adjusted observed variable. Represents observed variables Regarding key latent variables The estimated mixing coefficients, Representing key latent variables Corresponding pure child node The estimated mixing coefficients, Representing key latent variables The corresponding pure child node, Indicates key latent variables. Represents the set of key latent variables; Based on the observed variables and the adjusted observed variables, the adjusted observed variable pairs are obtained. ; Step 3: Using the cumulative unbounded criterion, determine the pairiness of each variable to be tested. If there is a synchronous instantaneous causal edge between them, then the causal direction is determined by the cumulative quantity orientation criterion, and a candidate synchronous edge set is obtained. Step 4: Redundant edges are removed from the candidate synchronous edge set to obtain the synchronous instantaneous causal structure of the current observation time step.

[0025] Furthermore, the criterion for determining the unbounded nature of the cumulative quantity in step three is as follows: If the pair of variables to be tested satisfies a linear combination relationship, then a cumulative no-boundary judgment expression is defined; if the pair of variables to be tested satisfies the cumulative no-boundary judgment expression, then it is determined that there is no synchronous instantaneous causal edge between the pair of observed variables; otherwise, it is determined that there is a synchronous instantaneous causal edge between the pair of observed variables. The expression for the linear combination relation is:

[0026]

[0027] In the formula, This indicates that the observed variables share the non-Gaussian latent components. Represents noise components. Indicates noise components, Represents the mixing coefficient. Represents the mixing coefficient. Represents the mixing coefficient. This represents the i-th variable to be tested in the current observation time step. This represents the j-th variable to be tested in the current observation time step; The expression for determining the non-boundary nature of the cumulative quantity is:

[0028] In the formula, Indicates higher-order cumulants; The cumulative quantity orientation criterion is as follows: The observed variable pair includes a first independent latent component and a second independent latent component; let the first independent latent component be the non-Gaussian latent component shared by the observed variable pair, and the second independent latent component be the component belonging only to the observed variable. Independent noise components; Based on the first and second independent latent components, the direction score is calculated; if the direction score is greater than zero, the synchronous instantaneous causal direction is determined to be... ;like Then the instantaneous causal direction is determined as follows: ; The expression for the direction score is:

[0029] In the formula, Indicates the direction score. Represents the mixing coefficient. Represents the mixing coefficient. Indicates the first independent potential component. This represents the second independent potential component.

[0030] Furthermore, the process of constructing the path separation set is as follows: Based on the synchronous instantaneous causal structure and the temporal causal transmission structure, an auxiliary causal graph is constructed, expressed as follows:

[0031] In the formula, This represents an auxiliary cause-effect graph. This represents the set of nodes in an auxiliary cause-effect graph. Represents the set of edges in an auxiliary cause-effect graph; For any two distinct output boundary variables in the auxiliary cause-effect graph, a variable set is constructed using the d-separation rule. ,satisfy:

[0032] In the formula, This represents the output boundary variable in the auxiliary cause-effect graph. This represents the output boundary variables in the auxiliary cause-effect graph; By merging the variable sets in the auxiliary causal graph, we obtain the path separation set corresponding to the current observation time step, expressed as:

[0033] In the formula, This represents a path separation set.

[0034] Furthermore, the process of conditionalizing the standardized non-stationary undersampled time series data for the next observation time step is as follows: Based on the path separation set, construct a conditional distribution. ,in, This represents the observation vector for the next observation time step. This represents the path separation set at the current observation time step. Using the conditional distribution, the standardized non-stationary undersampled time series data for the next observation time step is conditionally processed to obtain the conditionally processed standardized non-stationary undersampled time series data.

[0035] Furthermore, the process of obtaining the global causal structure graph is as follows: Based on the time-delay causal structure, the synchronous instantaneous causal structure, and the time-series causal propagation structure of each observation time step, the global node set and the global edge set are obtained; The expression for the global node set is:

[0036] In the formula, Represents the global set of nodes. This represents the i-th observed variable at the r-th observation time step. This represents the output boundary variable pointing from the r-th observation time step to the next unobserved time period. Indicates the initial observation time step Input boundary variables, Represents the set of observed variable indices. Indicates the total number of observation time steps; The expression for the global edge set is:

[0037]

[0038] In the formula, Represents the global edge set. This represents the j-th variable in the unobserved input boundary at the r-th observation time step. Let the j-th observed variable be at the r-th observation time step. This represents the element in the i-th row and j-th column of the time-delay causal structure matrix. This represents the element in the i-th row and j-th column of the synchronous instantaneous causal structure matrix. This indicates that the i-th variable in the output boundary of the next unobserved time period is pointed to by the r-th observation time step. This represents the element in the i-th row and j-th column of the temporal causal transit structure matrix; Based on the global node set and the global edge set, the global causal structure graph is obtained, expressed as:

[0039] In the formula, This represents a global cause-effect structure graph.

[0040] Compared with existing technologies, the beneficial effects of this method are: This invention provides a method for identifying causal structures in non-stationary undersampled time-series data. First, non-stationary undersampled time-series data is acquired and standardized to construct a time-series causal association model, representing time-varying causal relationships in the non-stationary data. For the current observation time step, key latent variables from unobserved periods are identified, and the time-delay causal structure is recovered, solving the problem of missing time-series information caused by undersampling and restoring the causal influence of unobserved periods on the currently observed variables. By inferring the synchronous instantaneous causal structure and constructing the time-series causal transmission structure, joint modeling of causal relationships across observation time steps is achieved, representing the synchronous effects and causal transmission paths between observed variables. By constructing a path separation set and conditionally processing the data for the next observation time step, historical causal confounding across time steps is eliminated, ensuring the independence of recursive inference. Causal structure inference is performed on each observation time step, and the time-delay causal structure, synchronous instantaneous causal structure, and time-series causal transmission structure of each observation time step are concatenated. This enables the reconstruction of a global causal structure graph from non-stationary, undersampled time-series data with latent confounding, improving the accuracy of causal structure identification. Attached Figure Description

[0041] Figure 1 A flowchart illustrating the causal structure identification method for non-stationary undersampled time-series data proposed in this embodiment of the invention; Figure 2 A schematic diagram illustrating the true global causal structure of non-stationary undersampled time-series data in the longitudinal follow-up scenario of chronic kidney disease proposed in this embodiment of the invention; Figure 3 A schematic diagram illustrating the time-delay causal structure of non-stationary undersampled time-series data in the longitudinal follow-up scenario of chronic kidney disease proposed in this embodiment of the invention; Figure 4 A schematic diagram illustrating the synchronous instantaneous causal structure and temporal causal transit structure of non-stationary undersampled time-series data in the longitudinal follow-up scenario of chronic kidney disease proposed in this embodiment of the invention; Figure 5 This represents a global causal structure diagram of non-stationary undersampled time-series data in the longitudinal follow-up scenario of chronic kidney disease proposed in this embodiment of the invention. Detailed Implementation

[0042] The accompanying drawings are for illustrative purposes only and should not be construed as limiting the scope of this patent. To better illustrate this embodiment, some parts of the accompanying drawings may be omitted, enlarged, or reduced, and do not represent the actual dimensions; It is understandable to those skilled in the art that some well-known details may be omitted from the accompanying drawings.

[0043] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0044] The positional relationships depicted in the accompanying drawings are for illustrative purposes only and should not be construed as limiting this patent. Example 1 This embodiment proposes a causal structure identification method for non-stationary undersampled time series data, such as... Figure 1 The flowchart shown illustrates the method, and the method proposed in this embodiment generally includes the following steps: S1: Obtain non-stationary undersampled time series data containing several observed variables, and standardize it to obtain standardized non-stationary undersampled time series data; construct a time series causal relationship model based on the standardized non-stationary undersampled time series data. S2: For the current observation time step, based on the current standardized non-stationary undersampled time series data, identify the key latent variables in the unobserved periods of the time series causal relationship model; based on the key latent variables, restore the time-delay causal structure of each observed variable in the current observation time step; S3: Based on the time-delay causal structure, infer the synchronous instantaneous causal structure among the observed variables at the current observation time step; based on the synchronous instantaneous causal structure, construct the temporal causal transmission structure of each observed variable at the current observation time step pointing to the next unobserved period. S4: Based on the synchronous instantaneous causal structure and the temporal causal transmission structure, construct the path separation set; use the path separation set to conditionally process the standardized non-stationary undersampled time series data of the next observation time step; S5: Using the next observation time step as the current observation time step, and using the standardized non-stationary undersampled time series data after conditional processing as the current standardized non-stationary undersampled time series data, repeat steps S2 to S4 until all observation time steps are processed. Then, splice the time-delay causal structure, the synchronous instantaneous causal structure, and the time-series causal propagation structure of each observation time step along the time sequence to obtain the global causal structure graph.

[0045] The causal structure identification method for non-stationary undersampled time series data proposed in this embodiment can be applied to the whole-course causal relationship analysis of longitudinal follow-up data of chronic diseases. It can achieve unified modeling from local causal structure identification to global causal graph reconstruction in non-stationary, undersampled time series scenarios with hidden confusion. It can provide support for the whole-course mechanism analysis and intervention decision-making of chronic diseases such as chronic kidney disease (CKD) based on longitudinal follow-up data of chronic diseases.

[0046] In this embodiment, the non-stationary undersampled time series data containing several observed variables is standardized by applying zero mean and unit variance to all observed variables, so that each observed variable participates in the subsequent causal structure inference at a uniform scale.

[0047] In this embodiment, the observation time step refers to the actual time point at which the observation data is collected. Unobserved time periods refer to the time intervals between two adjacent observation time steps. These time periods contain several unobserved time points, i.e., potential time steps, where the actual state changes have not been directly observed. Because clinical follow-ups are usually limited by the length of medical visits and testing conditions, sampling time points are relatively sparse, and there are a large number of unobserved physiological evolution processes between adjacent observation time steps, forming undersampled data.

[0048] To uniformly characterize the unobserved time periods between adjacent observation time steps, in this embodiment, all unobserved time steps between any two adjacent observation time steps are abstracted into a single intermediate boundary time period. This intermediate boundary serves both as the subject of the unobserved time period that exerts a time-delayed causal influence on the observed variables of the current observation time step, and as the object on which the observed variables of the current observation time step pass on the temporal causality to the next unobserved time period.

[0049] In this embodiment, the fine-grained full-time index set represents all time points, including observed time steps and potential time steps.

[0050] In this embodiment, the process of constructing a time-series causal association model based on standardized non-stationary undersampled time-series data is as follows: Step 1: Set the subsampling rate of the non-stationary undersampled time series data to be... Set the observation time step set as ,in, Represents the set of observation time steps. This represents the R-th observation time step. Represents a fine-grained, full-time index set; For example, subsampling rate .

[0051] Step 2: Set each observation time step to include Then the nth observed variable, The expression for the observation vector at each observation time step is:

[0052] In the formula, Indicates the first The observation vector for each observation time step. Indicates the first One observed variable; Step 3: Take the unobserved period between any two adjacent observation time steps in the standardized non-stationary undersampled time series data as the intermediate boundary period, and set the current observation time step... The output boundary is denoted as The next observation time step The input boundary is denoted as The expression for defining the boundary equivalence relation is:

[0053] In the formula, Indicates the next observation time step The input boundary, Indicates the current observation time step The output boundary; By utilizing boundary equivalence relations, unobserved processes between any adjacent observation time steps can be compressed into a single intermediate boundary time period.

[0054] Step 4: Based on the boundary equivalence relation, construct a time-series causal association model, expressed as:

[0055]

[0056] In the formula, Indicates entering the current observation time step Unobserved input boundary, Indicates the current observation time step The output boundary points to the next unobserved time period. Indicates unobserved input boundary Observation vector at the current observation time step The time-delay causal structure, This represents the synchronous instantaneous causal structure within the current observation time step. This represents the temporal causal transit structure where each observed variable at the current observation time step points to the next unobserved time period. Represents the exogenous noise term. This represents the exogenous noise term.

[0057] Specifically, time-delay causal structure Simultaneous instantaneous causal structure Temporal causal transitive structure .

[0058] In this embodiment, a pure child node refers to an observed variable that is driven by only a single key latent variable and is not directly causally affected by other observed variables within the current observation time step. That is, a pure child node satisfies the following: its value change is determined solely by the key latent variable with the same index in the unobserved period, and it is not affected by the contemporaneous instantaneous causal effects of other observed variables within the current observation time step; furthermore, the pure child node does not have a contemporaneous instantaneous causal effect on other observed variables within the current observation time step. A non-pure child node refers to the remaining observed variables in the current observation time step other than pure child nodes, i.e., observed variables that may be jointly affected by multiple key latent variables or may be subject to the contemporaneous causal effects of other observed variables.

[0059] In this embodiment, the key latent variable is the input boundary of the current observation time step. Key latent variables are unobserved non-Gaussian latent variables that can have cross-variable effects on multiple observed variables. In other words, key latent variables are non-Gaussian latent variables that are not directly observed and are shared only by at least three observed variables. Key latent variables exist in the unobserved periods between adjacent observation time steps and are the carriers of the time-lag causal effects of the unobserved periods on the observed variables at the current observation time step. A key latent variable can simultaneously drive multiple observed variables, resulting in statistical correlations among these observed variables caused by the key latent variable. For example, in longitudinal follow-up data of chronic kidney disease, unobserved systemic inflammation levels can simultaneously affect multiple clinical observation indicators such as blood pressure, renal function indicators, and proteinuria; this unobserved inflammation level is a key latent variable.

[0060] In this embodiment, the process of identifying key latent variables in the unobserved periods of the time-series causal association model based on the current standardized non-stationary undersampled time-series data is as follows: Step 1: Based on the current standardized non-stationary undersampled time series data, calculate the mixing coefficients of each observed variable at the current observation time step. The expression is:

[0061]

[0062] In the formula, Indicates higher-order cumulants. This represents the i-th observed variable at the current observation time step. This represents the j-th observed variable at the current observation time step. Represents observed variables The estimated mixing coefficients between the shared single key latent variable and the mixed coefficients. Represents observed variables Estimated mixing coefficients between the variable and a shared single key latent variable; Step 2: Combine any three observed variables in the current observation time into a triplet. Based on the mixing coefficient, determine each triplet in the current observation time step using the single latent component criterion. Combine the observed variables corresponding to the triplets that satisfy the single latent component criterion as the observed variable group. Each observed variable in the observed variable group shares the same key latent variable in the unobserved period. The observed variable set is expressed as follows:

[0063] In the formula, Represents the group of observed variables. Indicates the current observation time step The a-th observed variable, Indicates the current observation time step The One observed variable, Indicates the current observation time step The c-th observed variable; Step 3: Using each observed variable in the observed variable group as a proxy variable, perform regression on the other two observed variables in the observed variable group respectively to form a residual triplet, the expression of which is:

[0064] In the formula, Represents the residual triple. Represents a proxy variable. Represents proxy variables For observed variables The regression residuals after regression. Represents proxy variables For observed variables The regression residuals after regression; Specifically, ,make . This represents the indices of the three observed variables in the triplet. This represents the indices of the two remaining observation variables after removing the candidate observation variable h.

[0065] Step 4: If the residual triples satisfy the single latent component criterion, it indicates that the proxy variable... The source of influence is only the key latent variable; determine the proxy variable. This represents the pure child node of the key latent variable corresponding to the triple; Step 5: Identify the key hidden variables corresponding to nodes determined to be pure child nodes. and pure child nodes By using the same index for association, the key latent variables are uniquely determined in the observation variable space through the index of the pure child nodes, thus completing the identification of key latent variables in the unobserved period in the time-series causal association model.

[0066] Specifically, regarding the observation time step If there exists a variable allocation estimate coefficient that satisfies the single latent component criterion for the observed variable group, then the observed variable group is affected by the common time lag of the only key latent variable in the unobserved period.

[0067] In this embodiment, the single hidden component criterion is: For the current observation time step Let i represent the observed variable. index, Indicates the removal of observed variables The index set of the remaining observed variables, and ; The core determination formula of the single hidden component criterion is:

[0068] In the formula, Indicates statistical independence. This represents a non-all-zero weight vector. This represents the coefficient vector obtained based on the mixed coefficient estimation. This represents all observed variables corresponding to index set J at the current observation time step. This represents the i-th observed variable in the current observation time step; When the core determination formula of the single latent component criterion is met, the corresponding variable group is determined to meet the single latent component criterion. The current variable group only shares a single key latent variable, and each observed variable in the variable group shares the same key latent variable in the unobserved period.

[0069] Specifically, the calculation of the coefficient vector obtained based on the mixed coefficient estimation is expressed as follows:

[0070] In the formula, Represents observed variables Estimated mixing coefficients between the shared single key latent variable and the variable itself.

[0071] In this embodiment, the process of recovering the time-delay causal structure of each observed variable at the current observation time step based on the key latent variables is as follows: Step 1: For all observation variable groups that satisfy the single latent component criterion at the current observation time step, detect the pure child nodes corresponding to each observation variable group, associate the corresponding key latent variables through the pure child nodes, and establish a key latent variable group for each newly identified key latent variable. The key latent variable group contains the pure child nodes and coefficient vectors corresponding to the key latent variable. Step 2: Iteratively determine the remaining observation variable group. If the current observation variable group and the existing key latent variable group have at least one common observation variable, and the mixing coefficient of the common observation variable is consistent with the coefficient vector in the key latent variable group, then determine that the current observation variable group and the key latent variable group correspond to the same key latent variable. Add other observation variables in the current observation variable group that are not yet included in the key latent variable group to the key latent variable group, and recalculate the coefficient vector corresponding to the key latent variable group. Step 3: Based on the key latent variable groupings and coefficient vectors, determine the time-delay causal edges of each key latent variable pointing to each observed variable within the key latent variable grouping. For observed variables not covered by any key latent variable groupings, only the self-causal diagonal terms of the observed variable are retained to obtain the time-delay causal structure of the current observation time step.

[0072] In this embodiment, the path separation set is a set of variables used to block temporal historical path dependencies and block cross-time step causal confounding. An auxiliary causal graph is constructed based on the synchronous instantaneous causal structure and temporal causal transmission structure of the current observation time step. The path separation set is accurately determined and constructed on the auxiliary causal graph using the d-separation rule. Then, based on the conditional distribution of the path separation set, the observation data at the next observation time is subjected to conditional processing to eliminate the confounding effects of historical temporal paths.

[0073] In this embodiment, the process of constructing the path separation set is as follows: Based on the synchronous instantaneous causal structure and the temporal causal transmission structure, an auxiliary causal graph is constructed, expressed as follows:

[0074] In the formula, This represents an auxiliary cause-effect graph. This represents the set of nodes in an auxiliary cause-effect graph. Represents the set of edges in an auxiliary cause-effect graph; The expression for the node set of the auxiliary cause-effect graph is:

[0075] In the formula, This represents the set of observed variables at the current observation time step. This represents the set of output boundary variables that point from the current observation time step to the next unobserved period.

[0076] For any two distinct output boundary variables in the auxiliary cause-effect graph, a variable set is constructed using the d-separation rule. ,satisfy:

[0077] In the formula, This represents the output boundary variable in the auxiliary cause-effect graph. This represents the output boundary variables in the auxiliary cause-effect graph; By merging the variable sets in the auxiliary causal graph, we obtain the path separation set corresponding to the current observation time step, expressed as:

[0078] In the formula, This represents a path separation set.

[0079] In this embodiment, the process of conditionalizing the standardized non-stationary undersampled time series data for the next observation time step is as follows: Based on the path separation set, construct a conditional distribution. ,in, This represents the observation vector for the next observation time step. This represents the path separation set at the current observation time step. Using the conditional distribution, the standardized non-stationary undersampled time series data for the next observation time step is conditionally processed to obtain the conditionally processed standardized non-stationary undersampled time series data.

[0080] In this embodiment, the process of obtaining the global causal structure graph is as follows: Based on the time-delay causal structure, the synchronous instantaneous causal structure, and the time-series causal propagation structure of each observation time step, the global node set and the global edge set are obtained; The expression for the global node set is:

[0081] In the formula, Represents the global set of nodes. This represents the i-th observed variable at the r-th observation time step. This represents the output boundary variable pointing from the r-th observation time step to the next unobserved time period. Indicates the initial observation time step Input boundary variables, Represents the set of observed variable indices. Indicates the total number of observation time steps; The expression for the global edge set is:

[0082]

[0083] In the formula, Represents the global edge set. This represents the j-th variable in the unobserved input boundary at the r-th observation time step. Let the j-th observed variable be at the r-th observation time step. This represents the element in the i-th row and j-th column of the time-delay causal structure matrix. This represents the element in the i-th row and j-th column of the synchronous instantaneous causal structure matrix. This indicates that the i-th variable in the output boundary of the next unobserved time period is pointed to by the r-th observation time step. This represents the element in the i-th row and j-th column of the temporal causal transit structure matrix; Based on the global node set and the global edge set, the global causal structure graph is obtained, expressed as:

[0084] In the formula, This represents a global cause-effect structure graph.

[0085] Specifically, for the initial observation time step Since there are no earlier historical local blocks, it is possible to directly... Restore the causal structure Simultaneous instantaneous causal structure and temporal causal transit structure .

[0086] Example 2 This embodiment specifically describes the process described in S3 of Embodiment 1, which involves inferring the synchronous instantaneous causal structure among the observed variables at the current observation time step based on the time-delay causal structure, and constructing the temporal causal transmission structure of the observed variables at the current observation time step pointing to the next unobserved period based on the synchronous instantaneous causal structure.

[0087] In this embodiment, when a pure child node is detected in an observed variable group that satisfies the single latent component criterion, the corresponding key latent variable with the same index is accepted as a new key latent variable, and the key latent variable grouping is initialized. Then, other observed variable groups are scanned. If there is variable overlap between the current observed variable group and an existing key latent variable group, and the mixing coefficient estimates on the overlapping variables are consistent, the remaining observed variables in that observed variable group are merged into the corresponding key latent variable group, and the coefficient vector of that key latent variable group is updated synchronously. This process is repeated until no new key latent variable groups can be merged into at the current observation time step.

[0088] For each recovered key latent variable group, add the edge pointing from the key latent variable to each observed variable within the group to the matrix. In the corresponding column, the unobserved input boundary is obtained. For the current observation time step The time-delay causal structure. For observed variables not covered by any key latent variable groupings, only the self-causal terms with the same index are retained, i.e., only the terms with the same index are retained. The diagonal elements.

[0089] In this embodiment, the process of inferring the synchronous instantaneous causal structure among the observed variables at the current observation time step based on the time-delay causal structure is as follows: Step 1: Based on the time-delay causal structure, extract the set of pure child nodes and the set of non-pure child nodes in the current observation time step, and set all synchronous instantaneous causal edges corresponding to each pure child node in the set of pure child nodes to zero; Step 2: Form an unordered variable pair from any two non-pure child nodes in the set of non-pure child nodes. ,in, This represents the i-th observed variable in the current observation time step. Let represent the j-th observed variable in the current observation time step; determine whether each unordered variable pair shares a key latent variable. If they share a key latent variable, perform one-sided reduction on the unordered variable pair that shares the key latent variable and construct the adjusted observed variable pair as the variable pair to be tested; if they do not share a key latent variable, then the unordered variable pair is used as the variable pair to be tested. The process of constructing the adjusted pairs of observed variables is as follows: Select As a one-sided reduction object, the adjusted observed variable is constructed as follows:

[0090] In the formula, Indicates the adjusted observed variable. Represents observed variables Regarding key latent variables The estimated mixing coefficients, Representing key latent variables Corresponding pure child node The estimated mixing coefficients, Representing key latent variables The corresponding pure child node, Indicates key latent variables. Represents the set of key latent variables; Based on the observed variables and the adjusted observed variables, the adjusted observed variable pairs are obtained. ; Step 3: Using the cumulative unbounded criterion, determine the pairiness of each variable to be tested. If there is a synchronous instantaneous causal edge between them, then the causal direction is determined by the cumulative quantity orientation criterion, and a candidate synchronous edge set is obtained. Step 4: Redundant edges are removed from the candidate synchronous edge set to obtain the synchronous instantaneous causal structure of the current observation time step.

[0091] In this embodiment, the criterion for determining the non-boundary nature of the cumulative amount in step three is: If the pair of variables to be tested satisfies a linear combination relationship, then a cumulative no-boundary judgment expression is defined; if the pair of variables to be tested satisfies the cumulative no-boundary judgment expression, then it is determined that there is no synchronous instantaneous causal edge between the pair of observed variables; otherwise, it is determined that there is a synchronous instantaneous causal edge between the pair of observed variables. The expression for the linear combination relation is:

[0092]

[0093] In the formula, This represents the shared non-Gaussian latent components of the observed variables (the noise is a latent variable of non-Gaussian noise). Indicates noise components, Indicates noise components, Represents the mixing coefficient. Represents the mixing coefficient. Represents the mixing coefficient. This represents the i-th variable to be tested in the current observation time step. This represents the j-th variable to be tested in the current observation time step; The expression for determining the non-boundary nature of the cumulative quantity is:

[0094] In the formula, Indicates higher-order cumulants; Among them, observed variables and of The expression for the order cumulant is:

[0095] The cumulative quantity orientation criterion is as follows: The observed variable pair includes a first independent latent component and a second independent latent component; let the first independent latent component be the non-Gaussian latent component shared by the observed variable pair, and the second independent latent component be the component belonging only to the observed variable. Independent noise components; The expression for the first independent potential component is:

[0096] In the formula, Indicates the first independent potential component. This represents the shared non-Gaussian latent components of the observed variables; The expression for the second independent latent component is:

[0097] In the formula, Indicates the second independent potential component. Indicates that it belongs only to the observed variable Independent noise components.

[0098] Based on the first and second independent latent components, the direction score is calculated; if the direction score is greater than zero, the synchronous instantaneous causal direction is determined to be... ;like Then the instantaneous causal direction is determined as follows: ; The expression for the direction score is:

[0099] In the formula, Indicates the direction score. Represents the mixing coefficient. Represents the mixing coefficient. Indicates the first independent potential component. This represents the second independent potential component.

[0100] In this embodiment, redundant edges are removed from the candidate synchronous edge set to obtain the synchronous instantaneous causal structure at the current observation time step, specifically as follows: If the dependency caused by a candidate edge can be completely explained by other directed paths within the same observation time step, then the candidate edge is considered redundant and is removed. Other directed path explanations indicate that between the two observed variables connected by the explained candidate edge, there exists a directed path passing through one or more other observed variables. Each edge on this path has been determined in the candidate contemporaneous edge set, and this directed path can completely transmit the causal effect corresponding to the explained candidate edge without generating additional information inconsistent with the edge's direction. After removing redundant edges, the contemporaneous instantaneous causal structure of the current observation time step is obtained. .

[0101] In this embodiment, based on the synchronous instantaneous causal structure, the temporal causal propagation structure that points each observed variable at the current observation time step to the next unobserved period is constructed as follows: By utilizing the self-causal rule and based on the synchronous instantaneous causal structure, the causal effects corresponding to each directed edge in the synchronous instantaneous causal structure are transmitted to the next unobserved time period. At the same time, the self-causal transitive terms pointing to themselves by each observed variable are retained, thus constructing a temporal causal transitive structure.

[0102] If the synchronous instantaneous causal structure is If a synchronous instantaneous causal edge exists, then the temporal causal transit structure is denoteed. This indicates that the causal effect of the same period is transmitted to the next unobserved period. Simultaneously, for all observed variables, the diagonal term from the causal relationship is retained. This characterizes the causal continuation of the state of each observed variable to the next unobserved period. Therefore, the temporal causal transmission structure... It also encodes the transmission of instantaneous causal effects within the current observation time step to the next unobserved boundary period, as well as the autoregressive continuation effects of each variable.

[0103] Example 3 In this embodiment, the causal structure identification method for non-stationary undersampled time series data proposed in Embodiment 1 is illustrated by way of example.

[0104] like Figure 2 The diagram illustrates the true global causal structure of non-stationary undersampled time-series data in a longitudinal follow-up scenario for chronic kidney disease. It assumes that each observation time step contains d=6 clinical follow-up variables, denoted as follows: For example, these can be sequentially correlated with estimated glomerular filtration rate, systolic blood pressure, urine albumin-to-creatinine ratio, serum creatinine, fasting blood glucose, and inflammation-related indicators. Since clinical follow-ups are typically sampled monthly, quarterly, or longer, while the actual disease progression continues between adjacent follow-ups, there are unobserved intermediate boundary periods between adjacent observation time steps. During the entire progression of chronic kidney disease (CKD), some causal effects are transmitted from unobserved periods to the current observation time step via intermediate boundaries. Simultaneously, there may be concurrent instantaneous causal effects and temporal transmission effects to the next unobserved period within the current observation time step, thus constituting a complete full-time causal structure under non-stationary undersampled time series.

[0105] like Figure 3 The diagram illustrates the time-delay causal structure of non-stationary undersampled time-series data in a longitudinal follow-up scenario for chronic kidney disease, using d=6 observed variables as an example. Let's assume that at the current observation time step... Below, after using the single latent component criterion and pure child node detection, two key latent variables were located, namely... and .in, for The pure child node is recovered. right The time-delayed causal effect; for The pure child node is recovered. right The time-delayed causal effect is obtained. Figure 3 Time-delay causal structure, i.e., matrix The non-zero support and corresponding coefficients. In the CKD context, key latent variables can be understood as potential disease progression drivers that are not directly observed between two adjacent follow-up visits but can jointly influence the evolution of multiple clinical indicators.

[0106] like Figure 4 The diagram illustrates the concurrent instantaneous causal structure and temporal causal transit structure of non-stationary undersampled time-series data in a longitudinal follow-up scenario for chronic kidney disease. It shows the causal structure after the time delay has been restored. Based on this, after performing one-sided reduction, cumulative boundary determination, direction determination, and redundant edge removal on non-pure child node variable pairs, the synchronous instantaneous causal structure within the current observation time step is further obtained. And according to the self-causal rule, a propagation structure is constructed from the current observation time step to the next unobserved boundary time step. In the longitudinal follow-up scenario of CKD, based on the identified but unobserved disease-driving factors, it is necessary to further distinguish the direct effects of various clinical indicators at the same follow-up time, and how these effects continue to be transmitted to the next unobserved disease stage. Figure 4 This corresponds to the complete local recovery process from the local incoming boundary structure to the instantaneous structure within the observation block, and then to the output boundary propagated structure.

[0107] like Figure 5 The diagram shows the global causal structure of non-stationary undersampled time-series data in a longitudinal follow-up scenario of chronic kidney disease, combined with... Figure 2 This diagram illustrates the true full-time causal structure of a non-stationary undersampled time series in a longitudinal follow-up scenario for CKD. Figure 3 This represents the time-delay causal structure of the unobserved input boundary to the current observation block, recovered using pure child node anchoring and the single hidden component criterion at the current observation time step. ; Figure 4 Indicates that in the known Based on this, the synchronous instantaneous causal structure within the current observation time step is restored through one-sided reduction, cumulative boundary-free determination, direction determination, and redundant edge removal. Furthermore, a temporal causal propagation structure is constructed, pointing from the current observation time step to the next unobserved boundary period. ; Figure 5 Represents path-based separation sets After conditionalizing the next observation time, the above local recovery steps are recursively iterated to stitch together all local causal structures step by step, ultimately forming a complete temporal global causal structure graph. This process realizes a unified technical workflow from local causal structure identification to global causal graph reconstruction in non-stationary subsampling scenarios, and can provide structured causal evidence for CKD full-course mechanism analysis and intervention decision support.

[0108] The embodiments described are merely examples to clearly illustrate the present invention and are not intended to limit the implementation of the invention. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively describe all possible implementations. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A method for identifying causal structures in non-stationary undersampled time-series data, characterized in that, Includes the following steps: S1: Obtain non-stationary undersampled time series data containing several observed variables, and standardize it to obtain standardized non-stationary undersampled time series data; construct a time series causal relationship model based on the standardized non-stationary undersampled time series data. S2: For the current observation time step, based on the current standardized non-stationary undersampled time series data, identify the key latent variables in the unobserved periods of the time series causal relationship model; based on the key latent variables, restore the time-delay causal structure of each observed variable in the current observation time step; S3: Based on the time-delay causal structure, infer the synchronous instantaneous causal structure among the observed variables at the current observation time step; based on the synchronous instantaneous causal structure, construct the temporal causal transmission structure of each observed variable at the current observation time step pointing to the next unobserved period. S4: Based on the synchronous instantaneous causal structure and the temporal causal transmission structure, construct the path separation set; use the path separation set to conditionally process the standardized non-stationary undersampled time series data of the next observation time step; S5: Using the next observation time step as the current observation time step, and using the standardized non-stationary undersampled time series data after conditional processing as the current standardized non-stationary undersampled time series data, repeat steps S2 to S4 until all observation time steps are processed. Then, splice the time-delay causal structure, the synchronous instantaneous causal structure, and the time-series causal propagation structure of each observation time step along the time sequence to obtain the global causal structure graph.

2. The causal structure identification method for non-stationary undersampled time series data according to claim 1, characterized in that, The process of constructing a time-series causal association model based on standardized non-stationary undersampled time-series data is as follows: Step 1: Set the subsampling rate of the non-stationary undersampled time series data to be... Set the observation time step set as ,in, Represents the set of observation time steps. This represents the R-th observation time step. Represents a fine-grained, full-time index set; Step 2: Set each observation time step to include Then the nth observed variable, The expression for the observation vector at each observation time step is: In the formula, Indicates the first The observation vector for each observation time step. Indicates the first One observed variable; Step 3: Take the unobserved period between any two adjacent observation time steps in the standardized non-stationary undersampled time series data as the intermediate boundary period, and set the current observation time step... The output boundary is denoted as The next observation time step The input boundary is denoted as The expression for defining the boundary equivalence relation is: In the formula, Indicates the next observation time step The input boundary, Indicates the current observation time step The output boundary; Step 4: Based on the boundary equivalence relation, construct a time-series causal association model, expressed as: In the formula, Indicates entering the current observation time step Unobserved input boundary, Indicates the current observation time step The output boundary points to the next unobserved time period. Indicates unobserved input boundary Observation vector at the current observation time step The time-delay causal structure, This represents the synchronous instantaneous causal structure within the current observation time step. This represents the temporal causal transit structure where each observed variable at the current observation time step points to the next unobserved time period. Represents the exogenous noise term. This represents the exogenous noise term.

3. The causal structure identification method for non-stationary undersampled time series data according to claim 1, characterized in that, Based on the time-series causal association model and the current standardized non-stationary undersampled time-series data, the process of identifying key latent variables in the unobserved periods of the time-series causal association model is as follows: Step 1: Based on the current standardized non-stationary undersampled time series data, calculate the mixing coefficients of each observed variable at the current observation time step. The expression is: In the formula, Indicates higher-order cumulants. This represents the i-th observed variable at the current observation time step. This represents the j-th observed variable at the current observation time step. Represents observed variables The estimated mixing coefficients between the shared single key latent variable and the mixed coefficients. Represents observed variables Estimated mixing coefficients between the variable and a shared single key latent variable; Step 2: Combine any three observed variables in the current observation time into a triplet. Based on the mixing coefficient, determine each triplet in the current observation time step using the single latent component criterion. Combine the observed variables corresponding to the triplets that satisfy the single latent component criterion as the observed variable group. Each observed variable in the observed variable group shares the same key latent variable in the unobserved period. The observed variable set is expressed as follows: In the formula, Represents the group of observed variables. Indicates the current observation time step The a-th observed variable, Indicates the current observation time step The One observed variable, Indicates the current observation time step The c-th observed variable; Step 3: Using each observed variable in the observed variable group as a proxy variable, perform regression on the other two observed variables in the observed variable group respectively to form a residual triplet, the expression of which is: In the formula, Represents the residual triple. Represents a proxy variable. Represents proxy variables For observed variables The regression residuals after regression. Represents proxy variables For observed variables The regression residuals after regression; Step 4: If the residual triples satisfy the single latent component criterion, it indicates that the proxy variable... The source of influence is only the key latent variable; determine the proxy variable. This represents the pure child node of the key latent variable corresponding to the triple; Step 5: Identify the key hidden variables corresponding to nodes determined to be pure child nodes. and pure child nodes By using the same index for association, the key latent variables are uniquely determined in the observation variable space through the index of the pure child nodes, thus completing the identification of key latent variables in the unobserved period in the time-series causal association model.

4. The causal structure identification method for non-stationary undersampled time series data according to claim 3, characterized in that, The single hidden component criterion is as follows: For the current observation time step Let i represent the observed variable. index, Indicates the removal of observed variables The index set of the remaining observed variables, and ; The core determination formula of the single hidden component criterion is: In the formula, Indicates statistical independence. This represents a non-all-zero weight vector. This represents the coefficient vector obtained based on the mixed coefficient estimation. This represents all observed variables corresponding to index set J at the current observation time step. This represents the i-th observed variable in the current observation time step; When the core determination formula of the single latent component criterion is met, the corresponding variable group is determined to meet the single latent component criterion. The current variable group only shares a single key latent variable, and each observed variable in the variable group shares the same key latent variable in the unobserved period.

5. The causal structure identification method for non-stationary undersampled time series data according to claim 3, characterized in that, Based on key latent variables, the process of recovering the time-delay causal structure of each observed variable at the current observation time step is as follows: Step 1: For all observation variable groups that satisfy the single latent component criterion at the current observation time step, detect the pure child nodes corresponding to each observation variable group, associate the corresponding key latent variables through the pure child nodes, and establish a key latent variable group for each newly identified key latent variable. The key latent variable group contains the pure child nodes and coefficient vectors corresponding to the key latent variable. Step 2: Iteratively determine the remaining observation variable group. If the current observation variable group and the existing key latent variable group have at least one common observation variable, and the mixing coefficient of the common observation variable is consistent with the coefficient vector in the key latent variable group, then determine that the current observation variable group and the key latent variable group correspond to the same key latent variable. Add other observation variables in the current observation variable group that are not yet included in the key latent variable group to the key latent variable group, and recalculate the coefficient vector corresponding to the key latent variable group. Step 3: Based on the key latent variable groupings and coefficient vectors, determine the time-delay causal edges of each key latent variable pointing to each observed variable within the key latent variable grouping. For observed variables not covered by any key latent variable groupings, only the self-causal diagonal terms of the observed variable are retained to obtain the time-delay causal structure of the current observation time step.

6. The causal structure identification method for non-stationary undersampled time series data according to claim 5, characterized in that, Based on the aforementioned time-delay causal structure, the process of inferring the synchronous instantaneous causal structure among the observed variables at the current observation time step is as follows: Step 1: Based on the time-delay causal structure, extract the set of pure child nodes and the set of non-pure child nodes in the current observation time step, and set all synchronous instantaneous causal edges corresponding to each pure child node in the set of pure child nodes to zero; Step 2: Form an unordered variable pair from any two non-pure child nodes in the set of non-pure child nodes. ,in, This represents the i-th observed variable in the current observation time step. Let represent the j-th observed variable in the current observation time step; determine whether each unordered variable pair shares a key latent variable. If they share a key latent variable, perform one-sided reduction on the unordered variable pair that shares the key latent variable and construct the adjusted observed variable pair as the variable pair to be tested; if they do not share a key latent variable, then the unordered variable pair is used as the variable pair to be tested. The process of constructing the adjusted pairs of observed variables is as follows: Select As a one-sided reduction object, the adjusted observed variable is constructed as follows: In the formula, Indicates the adjusted observed variable. Represents observed variables Regarding key latent variables The estimated mixing coefficients, Representing key latent variables Corresponding pure child node The estimated mixing coefficients, Representing key latent variables The corresponding pure child node, Indicates key latent variables. Represents the set of key latent variables; Based on the observed variables and the adjusted observed variables, the adjusted observed variable pairs are obtained. ; Step 3: Using the cumulative unbounded criterion, determine the pairiness of each variable to be tested. If there is a synchronous instantaneous causal edge between them, then the causal direction is determined by the cumulative quantity orientation criterion, and a candidate synchronous edge set is obtained. Step 4: Redundant edges are removed from the candidate synchronous edge set to obtain the synchronous instantaneous causal structure of the current observation time step.

7. The causal structure identification method for non-stationary undersampled time series data according to claim 6, characterized in that, The criterion for determining the unbounded nature of the cumulative quantity mentioned in step three is as follows: If the pairs of variables to be tested satisfy a linear combination relationship, then define the cumulant unbounded judgment expression; If the test variable pair satisfies the cumulative no-boundary judgment expression, then it is determined that there is no synchronous instantaneous causal edge between the observed variable pair; Otherwise, it is determined that there is a synchronous instantaneous causal edge between the observed variable pairs; The expression for the linear combination relation is: In the formula, This indicates that the observed variables share the non-Gaussian latent components. Indicates noise components, Indicates noise components, Represents the mixing coefficient. Represents the mixing coefficient. Represents the mixing coefficient. This represents the i-th variable to be tested in the current observation time step. This represents the j-th variable to be tested in the current observation time step; The expression for determining the non-boundary nature of the cumulative quantity is: In the formula, Indicates higher-order cumulants; The cumulative quantity orientation criterion is as follows: The observed variable pair includes a first independent latent component and a second independent latent component; let the first independent latent component be the non-Gaussian latent component shared by the observed variable pair, and the second independent latent component be the component belonging only to the observed variable. Independent noise components; Based on the first and second independent latent components, the direction score is calculated; if the direction score is greater than zero, the synchronous instantaneous causal direction is determined to be... ;like Then the instantaneous causal direction is determined as follows: ; The expression for the direction score is: In the formula, Indicates the direction score. Represents the mixing coefficient. Represents the mixing coefficient. Indicates the first independent potential component. This represents the second independent potential component.

8. The causal structure identification method for non-stationary undersampled time series data according to claim 1, characterized in that, The process of constructing path split sets is as follows: Based on the synchronous instantaneous causal structure and the temporal causal transmission structure, an auxiliary causal graph is constructed, expressed as follows: In the formula, This represents an auxiliary cause-effect graph. This represents the set of nodes in an auxiliary cause-effect graph. Represents the set of edges in an auxiliary cause-effect graph; For any two distinct output boundary variables in the auxiliary cause-effect graph, a variable set is constructed using the d-separation rule. ,satisfy: In the formula, This represents the output boundary variable in the auxiliary cause-effect graph. This represents the output boundary variables in the auxiliary cause-effect graph; By merging the variable sets in the auxiliary causal graph, we obtain the path separation set corresponding to the current observation time step, expressed as: In the formula, This represents a path separation set.

9. A method for identifying causal structures in non-stationary undersampled time-series data according to claim 1, characterized in that, The process of conditionalizing the standardized non-stationary undersampled time series data for the next observation time step is as follows: Based on the path separation set, construct a conditional distribution. ,in, This represents the observation vector for the next observation time step. This represents the path separation set at the current observation time step. Using the conditional distribution, the standardized non-stationary undersampled time series data for the next observation time step is conditionally processed to obtain the conditionally processed standardized non-stationary undersampled time series data.

10. The causal structure identification method for non-stationary undersampled time series data according to claim 1, characterized in that, The process of obtaining the global cause-effect structure graph is as follows: Based on the time-delay causal structure, the synchronous instantaneous causal structure, and the time-series causal propagation structure of each observation time step, the global node set and the global edge set are obtained; The expression for the global node set is: In the formula, Represents the global set of nodes. This represents the i-th observed variable at the r-th observation time step. This represents the output boundary variable pointing from the r-th observation time step to the next unobserved time period. Indicates the initial observation time step Input boundary variables, Represents the set of observed variable indices. Indicates the total number of observation time steps; The expression for the global edge set is: In the formula, Represents the global edge set. This represents the j-th variable in the unobserved input boundary at the r-th observation time step. Let the j-th observed variable be at the r-th observation time step. This represents the element in the i-th row and j-th column of the time-delay causal structure matrix. This represents the element in the i-th row and j-th column of the synchronous instantaneous causal structure matrix. This indicates that the i-th variable in the output boundary of the next unobserved time period is pointed to by the r-th observation time step. This represents the element in the i-th row and j-th column of the temporal causal transit structure matrix; Based on the global node set and the global edge set, the global causal structure graph is obtained, expressed as: In the formula, This represents a global cause-effect structure graph.

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