Multi-factor data prediction modeling method based on space-time-frequency dynamic fusion graph neural network

Abstract

The invention belongs to the technical field of high-dimensional multi-factor data prediction, and particularly relates to a multi-factor data prediction modeling method based on a space-time-frequency dynamic fusion graph neural network. A low-rank tensor decomposition method based on a Schatten capped p norm is used for missing data in the data set, and a threshold value is introduced to adjust punishment of the norm to a singular value; by introducing a sparse error term E, the model can separate and process an abnormal value from normal data; constructing an alternating direction multiplier parameter optimization method based on momentum variance cooperation, introducing a momentum item by using a momentum acceleration method, and accelerating convergence of a current batch optimization algorithm by using historical iteration information; according to the graph neural network method based on space-time-frequency dynamic fusion, an adjacent matrix is dynamically generated through learning node embedding. The method has the beneficial effects that firstly, low-rank tensor decomposition is taken as a core, the correlation of data in time and space is captured in combination with a Schatten capped p norm, and the convergence efficiency and the capability of adapting to dynamic change are improved;

Description

Technical Field

[0001] This invention belongs to the fields of intelligent traffic management, environmental meteorological monitoring, and high-dimensional multi-factor data prediction technology, specifically involving a multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network. Background Technology

[0002] Sequence data is a sequence of data points arranged chronologically, essentially reflecting the trend of one or more random variables changing over time. Given the complexity of nonlinear systems, sequence data often exhibits complex nonlinearity and randomness, making sequence data prediction more challenging. Furthermore, with the continuous development of sensor, information communication, and computer storage technologies, and the arrival of the big data era, massive amounts of data are generated across all industries. Over time, the amount of data generated in various fields is growing exponentially, and the types and content of this data are becoming increasingly diverse.

[0003] Data loss is a common issue in real-world sequence data. Due to environmental interference, sensor malfunctions, human error, network communication interruptions, and many other factors, data loss is unavoidable, and sequence data acquired by sensors often exhibits unstructured missing characteristics. This not only affects the completeness and accuracy of the data but can also lead to biases in model training. Because real-time data streams are irreversible, traditional data imputation methods often cannot effectively handle large-scale missing data, resulting in a cascading negative impact on subsequent analysis and modeling. Data loss patterns are mainly divided into two categories: random missing (RM) and non-random missing (NM). Random missing means that the missing variable is related to other non-missing variables. Non-random missing means that the missing variable is related to both other non-missing variables and itself; this dependency significantly increases the complexity of data repair.

[0004] Existing research has made some progress in specific scenarios, but many problems remain to be solved. Most current methods do not adequately explore the dynamic changes in sequences. Furthermore, the complex correlations between sequence data have not been fully explored. In addition, there is a lack of targeted processing methods for mixed scenarios with random and non-random missing values. Therefore, this invention focuses on the problem of missing sensor data in sequence data modeling and proposes a data missing imputation method based on low-rank tensor decomposition-Schattencappedp norm. Using low-rank tensor decomposition as the core and incorporating the Schhattencappedp norm, this method captures temporal and spatial correlations, enabling the model to adapt to dynamic changes in data, improving the accuracy of missing value imputation, fundamentally ensuring reliable data, and enhancing the model's ability to match real-world systems. Finally, experiments are conducted using a traffic dataset from California, USA, and the method is compared with two matrix factorization-based methods and two tensor decomposition-based methods. Experimental results show that the low-rank tensor decomposition-Schattencappedp norm data missing imputation module can achieve high-precision imputation under different missing rates, and its error and correlation indices are superior to other comparative models.

[0005] To address the high computational complexity encountered when processing large-scale data, traditional ADMM methods reduce the computational complexity of each iteration by introducing randomization techniques, achieving a linear convergence rate. However, these methods still have a convergence rate of O(1 / T) on general convex problems, compared to the O(1 / T) of accelerated batch processing algorithms. 2 There is a gap. To bridge this gap, this invention, based on sequence data imputation, leverages momentum acceleration techniques and SVRG variance reduction techniques to design an alternating direction multiplier parameter optimization method and corresponding momentum term update rules based on momentum-variance synergy. By calculating the full gradient once at each stage and using the corrected gradient estimate in subsequent iterations, variance is significantly reduced. The momentum term uses a weighted average of historical gradients to reduce oscillations in the optimization path and accelerate convergence. This invention designs two different momentum term update rules, applicable to strongly convex and generally convex problems respectively. These rules can be used not only in specific environments but also integrated with other machine learning methods to improve model performance.

[0006] Sequence data often contains complex dynamic dependencies that traditional linear methods struggle to model effectively. To improve the predictive performance of high-dimensional, multivariate sequence data, it is necessary to learn or construct these dependencies. Vector Autoregression (VAR) models, as a baseline method, extend univariate autoregression (AR) models to allow interactions between multiple variables, providing a basic framework for sequence prediction. To capture nonlinear relationships, recurrent neural network models such as Long Short-Term Memory (LSTM) networks and Gated Recurrent Units (GRUs) are widely used in nonlinear sequence prediction due to their effective modeling of long-term dependencies. Convolutional Neural Networks (CNNs), through the combination of convolutional computation and deep neural structures, effectively extract local features and capture spatial dependencies in data. Temporal Convolutional Networks (TCNs), as a variant of CNNs, further enhance the ability to handle temporal relationships by introducing techniques such as causal convolution, dilated convolution, and residual connections, avoiding information leakage and becoming a research hotspot in the field of sequence prediction.

[0007] The Transformer model, with its self-attention mechanism, has gradually been introduced into the field of sequence prediction. While the Transformer excels at capturing the temporal correlation of sequence data such as traffic flow, its performance still faces challenges when processing data with dynamic periodic or aperiodic patterns. With the rise of graph learning techniques, data-driven sequence data prediction models have achieved new breakthroughs in the spatiotemporal dimensions. Graph Convolutional Networks (GCNs) extend convolution operations to graph-structured data through spatial and spectral domain convolutions, providing new approaches to handling complex spatiotemporal relationships.

[0008] This paper designs a multi-factor data prediction modeling method based on a spatiotemporal-frequency dynamic fusion graph neural network. In high-dimensional multi-factor sequences, each variable can be regarded as a node in a graph, and the hidden dependencies between variables can be represented by the edges of the graph. Due to the dynamic nature of high-dimensional multi-factor data, these dependencies change over time. A spatial dynamic graph neural network is used to capture this dynamic change. A time series modeling method based on frequency domain analysis and temporal convolution (Time Convolution and Fast Convolution 2D, TCN-FC2) is designed. By identifying and modeling periodic patterns and effectively capturing long-term dependencies, a deeper understanding of the dynamics and dependencies of the data can be achieved, thus providing a more accurate basis for prediction. The spatial features extracted by the spatiotemporal-frequency dynamic fusion graph neural network and the temporal features extracted by the fast two-dimensional convolutional network (frequency domain analysis) are concatenated, and the final result is output through a fully connected layer. This improves the performance of high-dimensional multi-factor collaborative prediction and enhances the ability to capture the dynamics, dependencies, and periodic features of time series data. Summary of the Invention

[0009] The purpose of this invention is to provide a multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network, which can effectively solve problems such as missing sensor data and uncertain correlation of multi-factor variables in multi-factor data prediction.

[0010] The technical solution of this invention is as follows: A multi-factor data prediction modeling method based on spatiotemporal-frequency dynamic fusion graph neural network. First, for missing data in the dataset, a low-rank tensor decomposition method based on the Schatten cappedp norm is used to fill in the missing data, utilizing the low-rank structure of the data. Combined with the Schatten cappedp norm, a threshold is introduced to adjust the norm's penalty for singular values. Second, by introducing a sparse error term E, the model can separate and process outliers from normal data. Third, an alternating direction multiplier parameter optimization method based on momentum-variance collaboration is constructed, utilizing a momentum acceleration method, introducing a momentum term, and using historical iteration information to accelerate the convergence of the current batch optimization algorithm. Finally, based on the spatiotemporal-frequency dynamic fusion graph neural network method, an adjacency matrix is ​​dynamically generated by learning node embeddings.

[0011] Includes the following steps:

[0012] Step 1: Construct a low-rank tensor decomposition method based on the Schatten cappedp norm;

[0013] Step 2: Construct a robust data missing imputation method for low-rank tensor decomposition;

[0014] Step 3: Construct an alternating direction multiplier parameter optimization method based on momentum-variance coordination;

[0015] Step 4: Temporal modeling based on spatial dynamic graph neural networks;

[0016] Step 5: Frequency domain analysis and temporal convolution are the two core isomorphic modules;

[0017] Step 6: Multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network.

[0018] Step 1 includes introducing the Schatten cappedp norm, which adjusts the penalty for singular values ​​by introducing a threshold.

[0019] For tensors Schatten cappedp norm extended to:

[0020]

[0021] Where τ is a threshold used to filter out singular values, X (k) Let X be the matrix obtained by expanding tensor X along the k-th mode, where k is the weight parameter, satisfying...

[0022] The objective function of the model is to minimize the Schatten cappedp norm of the tensor, while ensuring that the completed tensor is consistent with the observed partial data in terms of observation location.

[0023] The mathematical expression is:

[0024]

[0025] Where X is a third-order tensor representing the complete data; Y is a partially observed tensor containing missing values; P Ω (·) is the orthogonal projection operator on the observation set Ω; It is the Schatten cappedp norm of the tensor; the Schatten cappedp norm of the tensor is defined as the weighted sum of the Schattenp norms of each expanded matrix of the tensor, and the p-th power of the minimum value after comparing each singular value with the threshold τ.

[0026]

[0027] Where, σ i (X(k)) are the singular values ​​of the k-th expansion matrix of tensor X; τ is the threshold used to filter singular values; p∈(0,1] is a parameter that controls the strength of nonconvexity; α k It is the weight parameter of the k-th expanded matrix.

[0028] Step 2 includes proposing a low-rank tensor decomposition-Schatten cappedp norm data missing imputation method that introduces a sparse error term E for modeling and handling random outliers.

[0029] Step 3 includes variance reduction, which improves the convergence speed of the algorithm by reducing the variance of the gradient estimation, and is achieved by using SVRG gradient estimation.

[0030] Step 4 includes dynamically generating an adjacency matrix by learning node embeddings. Each node is assigned two embedding vectors, representing the features of the source node and the target node, respectively. For each time slice t, two vectors Θ and Ψ of length d are generated, where all elements are randomly initialized learnable parameters. Then, Θ is calculated. T Ψ serves as the initial adjacency matrix for a time slice, the values ​​of which are optimized through training.

[0031] Step 5 involves converting the one-dimensional sequence after FFT transformation into a set of two-dimensional tensors based on multiple periods, thereby extending the analysis of time-varying changes to a two-dimensional space. Through this transformation, the changes within the period are embedded into the columns of the two-dimensional tensor, and the changes between the periods are embedded into the rows. The two-dimensional change pattern of the sequence is modeled using two-dimensional convolution kernels.

[0032] Step 6 includes concatenating the spatial features extracted by the spatial dynamic graph neural network and the temporal features extracted by the two core isomorphic modules of frequency domain analysis and temporal convolution, and outputting the result through a fully connected layer. In each layer, the output feature H of the dynamic graph isomorphic network... (l) The feature X output by the temporal convolution fast 2D convolution method (l+1) Interact to form a new feature representation Z (l+1) :

[0033] Z (l+1) =Concat(H (l) ,X (l+1) )

[0034] In the final layer, the feature representations are further aggregated into a global feature representation through a global pooling operation:

[0035] G (L) =GlobalPool(Z (L) )

[0036] Among them, Z (L) The fusion feature representation of the last layer of the formula,

[0037] Finally, the global feature representation G (L) Passed to the output layer:

[0038]

[0039] Where FC represents a fully connected layer; b is a bias term.

[0040] The beneficial effects of this invention are as follows: First, it uses low-rank tensor decomposition as the core, combined with Schatten capped p-norm to capture the temporal and spatial correlations of data, improving convergence efficiency and adaptability to dynamic changes. Momentum acceleration and variance reduction techniques are introduced to optimize parameters, and an alternating direction multiplier parameter optimization method based on momentum-variance synergy is designed to enhance the model's matching ability with the actual system. Using a spatial dynamic graph neural network as the basic framework, a frequency domain analysis and temporal convolution isomorphic module is built. An attention mechanism is used to fuse multi-scale features across time, space, and frequency dimensions, uncovering the intrinsic correlations between different variables, achieving accurate prediction of multi-factor data synergy, and significantly improving the model's convergence rate while reducing computational complexity. Attached Figure Description

[0041] Figure 1 A diagram illustrating tensor completion;

[0042] Figure 2 Schematic diagrams of the stochastic gradient method and the momentum acceleration method;

[0043] Figure 3 This is a schematic diagram of dynamic graph transformation;

[0044] Figure 4 This is a schematic diagram of the FC2 structure;

[0045] Figure 5 This is a schematic diagram of the TCN structure;

[0046] Figure 6 This is a structural diagram of the frequency domain analysis and temporal convolution isomorphic module;

[0047] Figure 7 A schematic diagram of the overall structure of a spatiotemporal frequency dynamic fusion graph neural network for multi-factor data prediction;

[0048] Figure 8 A visual illustration of traffic scenarios and sudden change data;

[0049] Figure 9 Example image of a weather scene;

[0050] Figure 10 The effect of imputing missing values ​​in the TD4 and TD8 datasets is shown in the figure.

[0051] Figure 11 To fill in the correlation matrix before and after;

[0052] Figure 12 A graph showing the changes in indicators for different prediction step sizes;

[0053] Figure 13 A diagram illustrating the structure of the data in a single image;

[0054] Figure 14 This is a visual diagram illustrating the two-graph structure of the data.

[0055] Figure 15 A bar chart showing the evaluation indicators for three different prediction step sizes for the data;

[0056] Figure 16 This is a comparison chart of predicted and actual air quality values. Detailed Implementation

[0057] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0058] This invention provides a multi-factor data prediction modeling method based on a spatiotemporal frequency dynamic fusion graph neural network. First, it uses a low-rank tensor decomposition method based on the Schatten cappedp norm to impute missing data in the dataset, leveraging the low-rank structure of the data. Combining the Schatten cappedp norm, a threshold (capped) is introduced to adjust the norm's penalty for outliers, providing a flexible method to balance the trade-off between the kernel norm and the rank function, enabling the model to adapt to different datasets and missing data situations. Second, high-dimensional multi-factor data may contain random outliers (such as abnormal data caused by sensor malfunctions or sudden events). These outliers can interfere with the low-rank recovery process, leading to inaccurate imputation results. By introducing a sparse error term E, the model can effectively separate and process outliers from normal data, thereby improving the model's robustness and applicability.

[0059] Secondly, an alternating direction multiplier parameter optimization method based on momentum-variance coordination is constructed. Utilizing a momentum acceleration method, a momentum term is introduced, leveraging historical iteration information to accelerate the convergence of the current batch optimization algorithm, helping it converge to the optimal solution more quickly. This invention designs two different momentum term update rules, using the SVRG (Stochastic Variance-Reduced Gradient) variance reduction technique to reduce the variance of gradient estimation and minimize severe oscillations in the optimization path. It is applicable to strongly convex and generally convex problems and can be integrated with other machine learning methods to improve model performance.

[0060] Finally, this invention proposes a graph neural network method based on spatiotemporal-frequency dynamic fusion, which dynamically generates an adjacency matrix by learning node embeddings. Through the interaction of these two embedding vectors, the similarity or association strength between nodes can be dynamically calculated, thereby constructing an adjacency matrix that reflects the dynamic characteristics of the sequence. Two core isomorphic modules, frequency domain analysis and temporal convolution, are designed. By identifying and modeling periodic patterns and effectively capturing long-term dependencies, a deeper understanding of the dynamics and dependencies of the data can be achieved, thus providing a more accurate basis for prediction. The spatial features extracted by the spatial dynamic graph neural network are fused and concatenated with the temporal features extracted by the two core isomorphic modules, and the final result is output through a fully connected layer.

[0061] This invention has the following characteristics:

[0062] To address the complex problem of both missing and outlier data, this invention proposes a low-rank tensor decomposition method based on the Schatten-capped p norm, which captures the correlation between different data modalities. By introducing a threshold (capped), the penalty of the norm on singular values ​​can be flexibly adjusted, preserving important information in the data and handling both missing and outlier values ​​while removing noise or unimportant details.

[0063] Second, relying on momentum acceleration techniques and SVRG variance reduction techniques, we designed an alternating direction multiplier parameter optimization method based on momentum-variance coordination and a corresponding momentum term update rule. This reduced the oscillation of the optimization path, improved the model convergence rate, and effectively expanded the applicability of the model to strongly convex and general convex problems.

[0064] III. A multi-factor data prediction modeling method based on a spatiotemporal-frequency dynamic fusion graph neural network is proposed. This method uses a spatial dynamic graph neural network as its basic framework and proposes two core isomorphic modules: frequency domain analysis and temporal convolution. It extracts and effectively fuses deep features of multi-factor data from different dimensions of time, space, and frequency, significantly improving the ability to represent potential associations and collaborative prediction performance of high-dimensional multi-factor data. This results in reduced computational complexity and a significantly improved convergence rate, making it better suited for tasks such as dynamic representation, potential pattern mining, and long-sequence prediction of high-dimensional multi-factor data in complex real-world scenarios. This invention uses a low-rank tensor decomposition-Schatten cappedp norm data missing filling method, significantly improving the filling accuracy of the missing data model. Through an alternating direction multiplier parameter optimization method based on momentum-variance collaboration, the oscillation of the optimization path is reduced, effectively improving the model's computational efficiency and robustness. The multi-factor data prediction modeling method using a spatiotemporal-frequency dynamic fusion graph neural network improves the collaborative prediction performance of high-dimensional multi-factor data and enhances the ability to capture the dynamic, dependent, and periodic features of time-series data.

[0065] The principles of the present invention will be described below with reference to the accompanying drawings and Embodiments 1, 2, and 3.

[0066] A multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network includes the following steps:

[0067] Step 1: Construct a low-rank tensor decomposition method based on the Schatten cappedp norm;

[0068] In tensors, low rank indicates the existence of similarities or correlations between different dimensions (such as time and space). The rank of a tensor is a measure of its complexity, similar to the rank of a matrix, but because tensors have multiple dimensions, their rank is more complex. Figure 1As shown. To simplify the problem, this invention proposes the Schatten capped p-norm, which adjusts the norm's penalty for singular values ​​by introducing a threshold (capped). The Schatten capped p-norm can flexibly adjust its parameters to adapt to different data characteristics and missing value patterns.

[0069] For tensors The Schatten cappedp norm can be extended to:

[0070]

[0071] Here, τ is a threshold used to filter out singular values. When τ is sufficiently large, the Schatten cappedp norm degenerates into the ordinary Schatten p norm; when τ is sufficiently small, the Schatten cappedp norm degenerates into a rank function; X (k) Let X be the matrix obtained by expanding tensor X along the k-th mode, where k is the weight parameter, satisfying...

[0072] Schatten capped p-norm, by introducing a threshold τ, provides a flexible way to balance the trade-off between the nuclear norm and the rank function, enabling the model to adapt to different datasets and missing data.

[0073] The missing data imputation model is built upon the Schatten cappedp norm of the tensor, which is a unified representation between the kernel norm and the rank function. The objective function of the model is to minimize the Schatten cappedp norm of the tensor while ensuring that the imputed tensor is consistent with the observed partial data in terms of observation location.

[0074] The mathematical expression is:

[0075]

[0076] Where X is a third-order tensor representing the complete data; Y is a partially observed tensor containing missing values; P Ω (·) is the orthogonal projection operator on the observation set Ω; It is the Schatten cappedp norm of the tensor.

[0077] The Schatten capped p norm of a tensor is defined as the p-th power of the minimum value after weighted summation of the Schatten p norms of each expanded matrix of the tensor and comparison of each singular value with a threshold τ.

[0078]

[0079] Where, σ i (X(k)) are the singular values ​​of the k-th expansion matrix of tensor X; τ is the threshold used to filter singular values; p∈(0,1] is a parameter that controls the strength of nonconvexity; α k It is the weight parameter of the k-th expanded matrix.

[0080] Step 2: Construct a robust data missing imputation method for low-rank tensor decomposition;

[0081] Low-rank tensor decomposition methods based on the Schatten capped p-norm are primarily used for missing data completion tasks. Introducing a non-convex norm allows for better adaptation to the diversity and complexity of the data. However, current models assume the absence of outliers in the observed data, which is often not true in practical applications. High-dimensional, multi-factor data may contain random outliers (such as those caused by sensor malfunctions or sudden events), which can interfere with the low-rank recovery process, leading to inaccurate completion results.

[0082] To overcome the aforementioned limitations, a low-rank tensor decomposition-Schatten cappedp norm data missing imputation method is proposed, which introduces a sparse error term E for modeling and handling random outliers. Specific improvements are as follows.

[0083] Suppose that the observed data Y can be decomposed into two parts:

[0084] Y = X + E

[0085] Where X is a low-rank tensor representing the normal pattern of the data, and E is a sparse error tensor representing random outliers.

[0086] By introducing E, the low-rank tensor decomposition model based on the Schatten cappedp norm can separate outliers from normal data, thereby improving the robustness of the recovery results.

[0087] Therefore, the model optimization problem is extended to:

[0088]

[0089] in, It is the Schatten-Cappedp norm of the tensor, used to model low-rank structures; ||E k ||1 is the L1 norm of the sparse error tensor, used to model sparse outliers; λ k It is a balancing parameter that controls the importance of sparse terms; X k P is the k-th modal expansion matrix of tensor X; Ω It is a projection operator used to process observation data Y.

[0090] By introducing ||Ek With just one term, both missing and outlier values ​​can be handled simultaneously. The sparse error term E is modeled using the L1 norm, which allows outliers to be effectively separated and handled, thereby improving the robustness and applicability of the model.

[0091] Specifically, we introduce auxiliary variable M and Lagrange multipliers T. k The problem is transformed into:

[0092]

[0093] in, <T k ,X k +E k -M> is a Lagrange multiplier term used to constrain X. k +E k =M; It is a penalty term used to ensure that the constraint is satisfied; ρ k It is a penalty parameter.

[0094] For each k, update X. k The subproblems are:

[0095]

[0096] For matrix Perform singular value decomposition (SVD) to obtain UΣV T .

[0097] Applying a soft thresholding operation to the singular value Σ, we obtain the optimal value by solving the following optimization problem:

[0098]

[0099] Using the properties of the Schatten-Cappedp norm, we can obtain X. k Update formula:

[0100]

[0101] Among them, Q k It is an operator that refolds a matrix into a tensor.

[0102] The subproblem of updating M is:

[0103]

[0104] This is a weighted average problem, and its closed-form solution is:

[0105]

[0106] The update results of all modes are combined by weighted averaging, taking into account the influence of Lagrange multipliers.

[0107] Update E k The subproblems are:

[0108]

[0109] This is a sparse constraint problem, solved using a soft thresholding operation:

[0110]

[0111] in,

[0112] By utilizing the properties of the L1 norm, outliers can be separated through soft thresholding.

[0113] Update Lagrange multipliers T k The formula is:

[0114]

[0115] Step 3: Construct an alternating direction multiplier parameter optimization method based on momentum-variance coordination;

[0116] In traditional gradient descent, if the current gradient is in the same direction as the previous gradient, it accelerates in that direction; the greater the accumulated momentum, the faster the speed. However, if the current gradient is in a different direction than the previous gradient, it indicates a drastic change in the function, causing it to decelerate and adjust its direction. This simple gradient descent method can lead to slow convergence in some cases, especially when the objective function has high curvature or regions with drastic gradient changes. Momentum acceleration addresses this problem by introducing an additional momentum term, whose update formula is:

[0117]

[0118] Where θ is the momentum weight, representing the contribution of historical information. Momentum term θ(x) k -x k-1 In fact, it is a weighted sum of the directions of the previous iteration, which helps the algorithm remember the previous update directions, thereby reducing oscillations and accelerating convergence.

[0119] Variance reduction techniques improve the convergence speed of algorithms by reducing the variance of gradient estimation, such as... Figure 2 As shown, SVRG is a representative method that significantly reduces variance by computing the full gradient once at each stage and using the corrected gradient estimate in subsequent iterations.

[0120] Variance reduction techniques are achieved through gradient estimation using SVRG. Specifically, at the beginning of each stage, the full gradient is calculated once:

[0121]

[0122] In each iteration k, a mini-batch I is randomly selected. k And calculate the corrected gradient estimate:

[0123]

[0124] in, At the current iteration point Mini-batch gradient estimation at the location; It is a deviation correction between mini-batch gradients and full gradients; The current iteration point gradient at; It is the average value of the previous stage. The gradient at that point.

[0125] This corrected gradient estimation Under expectation, it equals the full gradient. This significantly reduces the variance of gradient estimation. By using this low-variance gradient estimation, the algorithm can more accurately approximate the true gradient in each iteration, thereby reducing oscillations in the optimization path and accelerating convergence.

[0126] Step 4: Temporal modeling based on spatial dynamic graph neural networks;

[0127] The adjacency matrix is ​​dynamically generated by learning node embeddings. Each node is assigned two embedding vectors, representing the features of the source node and the target node, respectively.

[0128] For each time slice t, generate two vectors Θ and Ψ of length d, where all elements are randomly initialized learnable parameters. Then calculate Θ. T Ψ serves as the initial adjacency matrix for a time slice. The values ​​of these time slice adjacency matrices can be optimized through training. The specific formula is shown below:

[0129] A = Θ T ·Ψ

[0130] Where A is the adjacency matrix; Θ = [θ t,1 ,θ t,2 ,…,θ t,d ],Ψ=[ψ t,1 ,ψ t,2 ,…,ψ t,d ] represents a randomly initialized learnable parameter vector, which represents the source embedding and target embedding of the node, respectively.

[0131] To reduce computational complexity, for each time slice's adjacency matrix, only the top k elements with the largest weights are retained, and the weights of the remaining edges are set to zero.

[0132] idx,idy=argtopk(A[:,:])idx≠idy

[0133] A[-idx,-idy]=0

[0134] Here, argtopk() returns the indices of the k largest values ​​in the adjacency matrix A; idx and idy are the row and column indices of the k elements with the largest weights.

[0135] To achieve information flow between time steps in a dynamic graph, this invention establishes connections between different graphs by designing a transformation method for dynamic graph transitions on the dynamic graph set. Except for the first time slice, the same number of vertices are added to each time slice, representing the data of the graph in the previous time slice for each corresponding vertex. The vertex set then becomes {v (t,1) ,v (t,2) ,…,v (t,N) ,v (t-1,1) ,v (t-1,2) ,…,v (t-1,N) Connect the corresponding positions of the previous time point and the current time point with a directed edge, that is, add an edge from v. (t-1,n) Pointing to v (t,n) The edges of the graph, where n = 1, 2, ..., N. The set of graphs after information aggregation is passed to the next step, such as... Figure 3 As shown.

[0136] Dynamic graph isomorphic networks can be defined as:

[0137]

[0138] In the formula, This represents the feature representation of node v at time slice t in layer l. It is the feature representation of node v in the previous layer (l-1) at the current time slice t; It is the feature representation of node v in the previous time slice t-1 of the upper layer (l-1); MLP (l) Multilayer perceptron, used for feature transformation; e (l) These are learnable parameters used to adjust the weights of a node's own features; It is the normalized edge weight; N(v) is the set of neighboring nodes of node v;

[0139] Feature merging is performed using the following formula:

[0140]

[0141] Among them, H(l) The global feature representation of layer l is obtained by concatenating features from different time steps.

[0142] The adjacency matrix is ​​in the form of:

[0143]

[0144] In the formula, H l-1 H represents the output tensor of the l-th graph isomorphic network layer; l-1 [t1:t T-1 [] indicates data alignment starting from the second time slice; pass The normalization method is used, and D is the degree matrix of A, which is the matrix of the number of edges connected to the node.

[0145] To address the issues of information loss and computational efficiency in traditional Graph Neural Networks (GNNs) when processing multivariate sequential data, convolutional operations are used to cluster nodes into a new node set, while simultaneously extracting temporal features.

[0146]

[0147] Among them, X (l+1) It is the feature representation of the (l+1)th layer; weight(N) l+1 ,j) represents the convolution kernel weights; * represents the two-dimensional convolution operation; bias(N) l+1 ) is the bias term; N l and N l+1 These represent the number of nodes before and after pooling, respectively.

[0148] Adjacency matrix update:

[0149] A (l+1) =M (l) A (l) (M (l) ) T

[0150] Among them, M (l) The allocation matrix has the shape [N] l+1 N l Generated through convolutional weights; A (l) This is the adjacency matrix of the current layer.

[0151] This update mechanism allows the graph structure to dynamically adapt to changes in the sequence while maintaining the graph's isomorphic properties.

[0152] Step 5: Frequency domain analysis and temporal convolution are the two core isomorphic modules;

[0153] The Fast Fourier Transform (FFT) can transform a sequence from the time domain to the frequency domain, revealing the intensity of different frequency components. By selecting the top K frequencies with the largest amplitudes, the most significant periodic patterns in the sequence can be identified. Converting a one-dimensional sequence after FFT transformation into a set of two-dimensional tensors based on multiple periods extends the analysis of time variations to two-dimensional space. Through this transformation, variations within a period can be embedded into the columns of the two-dimensional tensor, and variations between periods can be embedded into the rows, such as... Figure 4 The two-dimensional variation pattern of a sequence can be efficiently modeled using a two-dimensional convolution kernel. The specific formula is as follows:

[0154]

[0155] in, It is the transformed two-dimensional tensor; Padding(X) zero-padding the sequence to ensure divisibility by p. i and f i ; Rearrange the sequence into a two-dimensional tensor, where p i It is the number of rows (cycle length), f i It is the number of columns (the number of points within a period).

[0156] According to the following formula, multi-scale features are extracted through two-dimensional convolution to capture changes within the period and changes during the period.

[0157]

[0158] in, It is the feature tensor after two-dimensional convolution; Inception(·) is the Inception module, which contains multiple convolution kernels of different sizes and can capture local and global features simultaneously.

[0159] TCN (Temporal Convolutional Network) effectively captures long-range dependencies in sequences through causal convolution and dilated convolution, such as... Figure 5 As shown.

[0160] Suppose the input sequence is x = [x1, x2, ..., x... T The output of a causal convolution with kernel size k can be expressed as:

[0161]

[0162] Where F(s) is the output after convolution; f is the convolution kernel;

[0163] To maintain consistent input and output sequence lengths, TCN adds zero padding after each convolutional layer.

[0164] Dilated convolution expands the receptive field of the convolution kernel by introducing gaps, thus enabling it to capture a wider range of dependencies. The core idea of ​​dilated convolution is to skip some time steps within the kernel, rather than acting continuously at each time step.

[0165] Suppose the input sequence is x = [x1, x2, ..., x... T The kernel size is k, the dilation factor is d, and the output of the dilated convolution can be expressed as:

[0166]

[0167] When d=1, dilated convolution degenerates into ordinary convolution. This is achieved by exponentially increasing the dilation rate at network depth (e.g., d=2). i TCN can achieve an exponentially growing receptive field, thereby capturing longer historical information.

[0168] Residual connections are used to address the vanishing gradient problem during deep network training and allow the network to learn more complex features, as shown in the following formula:

[0169] O T =Activation(x+F(x)) x is the input; F(x) is the output of the network layer; O is the output after residual connection; Activation is the activation function, usually ReLU.

[0170] The model's representational power is enhanced by fusing the frequency domain features extracted by fast 2D convolution with the temporal features of TCN. An attention mechanism is selected to dynamically combine the two types of features to extract locally important information. Before fusion, the features from the fast 2D convolution need to be dimensionally aligned.

[0171]

[0172] Where, α i The weight of the i-th cycle is obtained by normalizing the amplitude A using Softmax; Trunc(·) truncates the padded portion, restoring the original sequence length T; Reshape 1,pi×fi (·) Rearranges a two-dimensional tensor into a one-dimensional tensor; X 1D It is a one-dimensional characteristic matrix.

[0173]

[0174] O F =Concat(O T (Attention Score × X) 1D ))

[0175] By fusing features extracted in two different ways, such as Figure 6 As shown, it can not only capture changes within and during the cycle, but also learn complex patterns and long-term dependencies in sequence data.

[0176] Step 6: A multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network;

[0177] The spatial features extracted by the spatial dynamic graph neural network and the temporal features extracted by the two core isomorphic modules of frequency domain analysis and temporal convolution are concatenated, and the result is output through a fully connected layer, such as... Figure 7 As shown.

[0178] In each layer, the output feature H of the dynamic graph isomorphic network (l) The feature X output by the temporal convolution fast 2D convolution method (l+1) Interact to form a new feature representation Z (l+1) :

[0179] Z (l+1) =Concat(H (l) ,X (l+1) )

[0180] In the final layer, the model further aggregates the feature representations into a global feature representation through a global pooling operation:

[0181] G (L) =GlobalPool(Z (L) )

[0182] Among them, Z (L) The fusion feature representation of the last layer of the formula.

[0183] Finally, the global feature representation G (L) Passed to the output layer:

[0184]

[0185] Where FC represents a fully connected layer; b is a bias term.

[0186] Example:

[0187] This invention was tested on two challenging high-dimensional, multi-factor traffic datasets and meteorological data. Figure 7-8The example images show traffic and weather scenarios. Traffic data includes the publicly available traffic flow dataset from the California Department of Transportation Performance Measurement System (PeMS), recording traffic data from the California Region 4 Highway System (TD4). This data contains three features: traffic flow, average speed, and average occupancy. Each feature dimension contains 288 time steps, covering 307 monitoring stations. Therefore, the data tensor shape is [3,288,307], with a total of 265,248 data points.

[0188] Data 2 includes a public traffic flow dataset provided by the California Department of Transportation Performance Measurement System (PeMS), recording data from eight highways (TD8) in San Bernardino. This data contains three features: traffic flow, average speed, and average occupancy. Each feature dimension contains 288 time steps, covering 170 monitoring stations. Therefore, the data tensor shape is [3, 288, 170], with a total of 146,880 data points.

[0189] Data Set 3 contains air quality and pollutant data for Beijing from December 1 to December 31, 2019, covering 28 air quality monitoring stations. The dataset features include air quality index (AQI), carbon monoxide (CO) concentration, and PM2.5 concentration, sampled every 30 minutes, totaling 41,664 data points. The correlation of the samples in Data Set 3 is shown in the figure.

[0190] Example 1:

[0191] This invention first conducts experiments on two traffic datasets to verify the imputation effect of high-dimensional multi-factor datasets with different missing rates. This experiment compares the imputation effect with two matrix factorization-based and two tensor factorization-based methods, using BTMF, BGCP, HalRTC, and LRTC-TNN as comparison models. Three factors were imputed, and the imputation effect was examined under random and non-random missing rate conditions of 20% and 40%, respectively. Data with missing rates exceeding 40% may significantly damage the data's inherent properties, thereby reducing its usability and reference value.

[0192] Table 1 Comparison of Filler Effects with a 20% Missing Rate Table 1 Comparison of Filler Effects with 20% Missing Rate

[0193]

[0194] Experimental results show that the missing mechanism has a significant impact on the imputation effect. Compared with random missing, the methods under non-random missing conditions have larger errors and lower correlations.

[0195] Table 2. Filling effect at a missing rate of 40%

[0196]

[0197] The proposed Low-Rank Tensor Decomposition-Schatten Cappedp Norm Data Missing Imputation (RLRO) method outperforms the comparative models at both 20% and 40% missing rates. RLRO shows the best performance on MAE, RMSE, SMAPE, and CORR metrics, indicating that its imputed data more closely approximates the true values. Although all models experience a performance degradation at high missing rates, RLRO maintains its best performance, demonstrating stronger robustness.

[0198] To address the issue of different traffic parameters having their own data characteristics, the imputation effectiveness of the three key variables was evaluated separately. Furthermore, by setting two gradients—20% (moderate missing) and 40% (severe missing)—the robustness of the imputation method under different levels of missing data can be comprehensively examined.

[0199] Table 3. Imputation effect of single factors under random and non-random missing values.

[0200]

[0201] Table 3 shows that the overall performance of data imputation is quite good under the random missing data mechanism. Especially for traffic flow data, the correlation coefficient (CORR) is close to 0.98, reflecting a high degree of consistency between the imputed data and the original real data. Even with the increase in the missing rate, the imputation effect of individual variables remains stable, without significant error accumulation. However, the average speed data, due to its small numerical range and significant differences from the other two variables, may interfere with the overall effect during the imputation process. Under the non-random missing data mechanism, the imputation effect is slightly lower than that under the random missing data mechanism. This is because non-random missing data typically manifests as consecutive missing data blocks, similar to long-term data interruptions caused by sensor malfunctions in real-world scenarios. In this case, the imputation module is limited in learning the temporal dependencies of the data due to the lack of key feature information, thus affecting the accuracy of the imputation.

[0202] Figure 10The imputation performance of the TD4 and TD8 datasets with missing values ​​of 20% and 40% under both random and non-random missing conditions is shown. Left (a) shows the imputation performance of the TD4 dataset with 20% random missing, and (b) shows the imputation performance of the TD4 dataset with 40% random missing. It can be seen that as the missing value increases from moderate to severe, the model's imputation performance does not decrease significantly. (c) shows the imputation performance of the TD4 dataset with 20% non-random missing, and (d) shows the imputation performance of the TD4 dataset with 40% non-random missing. The visualization clearly shows that the prediction performance for random missing values ​​is more accurate than that for non-random missing values. The right figure shows a comparison of the imputation performance of the TD8 dataset under different missing conditions. (a) and (b) present the imputation results with random missing values ​​of 20% and 40%, respectively. The visualization shows that although the missing value doubles, the model's imputation accuracy remains stable, demonstrating the algorithm's robustness to random missing values. In contrast, (c) and (d) show the performance of non-random missing data (simulating continuous data block missing data due to sensor failure) at the same missing rate, and it can be observed that the imputation effect is significantly weaker than that of random missing data scenarios.

[0203] Figure 11 The changes in feature correlations before and after data imputation are visually presented through a correlation matrix comparison. Subplots (a) and (c) are correlation heatmaps before imputation, while (b) and (d) are correlation heatmaps after imputation. The visualization shows that after processing with the Robust Low-Rank Tensor Optimization (RLRO) method, the average correlation coefficient among traffic flow, average speed, and occupancy rate increased from 0.44 before imputation to 0.6. The correlation coefficient between occupancy rate and average speed showed a significant change, shifting from a weak positive correlation (r = 0.20) to a strong negative correlation (r = -0.83), which is consistent with the basic laws of traffic flow theory. This indicates that missing values ​​in the original data may have masked the inherent spatiotemporal coupling relationships between traffic parameters. Furthermore, high-quality imputation not only restored data integrity but also revealed deep correlations that were previously obscured by noise and missing values ​​by mining potential patterns among multidimensional features.

[0204] To verify the performance of the spatiotemporal-frequency dynamic graph neural network multi-factor data prediction model (STFDG), the designed model was compared with other basic time series prediction models, as shown in Table 4, to verify its performance in basic multivariate prediction tasks. The comparison models were LSTM, GRU, Bi-LSTM, and Bi-GRU. To verify the spatiotemporal-frequency fusion adaptability and prediction performance of the STFDG model under different time lengths, the designed model was compared with other spatiotemporal fusion deep learning models, as shown in Tables 5-7. The comparison models used were STGCN, ASTGCN, TGCN, DCRNN, and LSGCN.

[0205] Table 4 Comparison Results of Basic Models

[0206]

[0207] Specifically, on the TD4 dataset, STFDG reduced the MAE by 22.2% and the RMSE by 18.3% compared to the best-performing comparison model, Bi-LSTM. On the TD8 dataset, STFDG reduced the MAE by 26.3% and the RMSE by 31.8% compared to the best-performing comparison model, Bi-LSTM. This demonstrates the superiority of STFDG in basic multivariate prediction tasks.

[0208] Table 5. Evaluation results of each model with a prediction length of 15 minutes.

[0209]

[0210] With a prediction step size of 15 minutes, the STFDG model performed exceptionally well on both TD4 and TD8 datasets. On the TD4 dataset, the STFDG model demonstrated performance comparable to or better than the best-performing model with a prediction step size of 15 minutes. On the TD8 dataset, STFDG showed a 2.5% reduction in MAE, a 3.1% reduction in RMSE, and a 1.8% reduction in SMAPE compared to the best-performing comparative model TGCN. STFDG's RMSE was 2.47, nearly identical to the LSGCN model with the lowest RMSE, and its evaluation metrics were 5.7%, 6.8%, and 7.6% lower than those of STGCN.

[0211] Table 6 Evaluation results of each model with a prediction length of 30 min.

[0212]

[0213] With a prediction step size of 30 minutes, in the TD4 experiment, STFDG reduced MAE, RMSE, and SMAPE by 6.2%, 6.7%, and 5.6%, respectively, compared to the relatively better-performing TGCN. In the dataset 2 experiment, STFDG reduced MAE and SMAPE by 3.9% and 6.8%, respectively, compared to the best-performing LSGCN model, while the RMSE was almost the same. Although it was comparable to LSGCN in RMSE in TD8, it outperformed LSGCN in MAE and SMAPE. These results indicate that STFDG outperforms the best-performing LSGCN and TGCD models in both prediction accuracy and precision.

[0214] Table 7 Evaluation results of each model with a prediction length of 45 min.

[0215]

[0216] With a prediction step size of 45 minutes, in the TD4 experiment, compared with the best-performing LSGCN model, the MAE, RMSE, and SMAPE decreased by 8.3%, 4.6%, and 6.9%, respectively. In the TD8 experiment, the STFDG model reduced MAE, RMSE, and SMAPE by 6%, 4%, and 2% compared to LSGCN, and by 10.8%, 11.1%, and 11.1% compared to the STGCN model, respectively. STFDG maintained a leading advantage in all three metrics: MAE, RMSE, and SMAPE.

[0217] Table 8 Evaluation results of each model with a prediction length of 60 min.

[0218]

[0219] With a prediction step size of 60 minutes, in the TD4 experiment, compared with the best-performing LSGCN model, the MAE, RMSE, and SMAPE decreased by 7.6%, 3.7%, and 5.7%, respectively. In the TD8 experiment, the STFDG model reduced the MAE by 6.5%, RMSE by 3%, and SMAPE by 5% compared to LSGCN. The advantages became more and more obvious as the prediction step size increased, indicating that the model performed well in balancing prediction length and prediction accuracy. Figure 12 The evaluation metrics in Tables 5-8 were visualized. In the figures, the horizontal axis represents different prediction step sizes, and the vertical axis corresponds to the three evaluation metrics. The first column shows the visualization results for dataset 1, and the second column shows the corresponding results for dataset 2. Through these visualizations, it is clear that even with continuously increasing prediction step sizes, the model proposed in this invention still outperforms other classic spatiotemporal learning models in all evaluation metrics, demonstrating a significant leading advantage.

[0220] Figure 13 and Figure 14 Two graph structures are demonstrated: a correlation graph and a graph generated by STFDG. The correlation graph is obtained by calculating the time-series correlation of traffic flow between stations, revealing potential dynamic relationships between stations. The graph generated by STFDG is the optimal graph structure learned by the algorithm. It integrates spatial and temporal information from the data, aiming to more accurately reflect the complex dependencies between stations. The graph structure generated by STFDG not only retains the original graph structure with high correlations, but also discovers new correlations and removes old ones while maintaining sparsity.

[0221] Example 2:

[0222] With the increasing prominence of global environmental issues, air quality forecasting, as a key technology, is leading the innovation of environmental management and public health protection. In the field of weather pollution, the application of high-dimensional, multi-factor data, combined with internet technology, big data analytics, and cloud computing, makes environmental monitoring more intelligent, enabling real-time assessment of pollution levels and the development of control strategies. By integrating multi-dimensional information such as meteorological data, pollutant emission data, and geographic information, air quality forecasting utilizes multi-factor analysis and modeling to predict changes in air quality in advance, optimize resource management and emergency response measures, and provide strong decision support for governments, businesses, and the public.

[0223] Air quality forecasting systems utilize sensor networks to collect multi-source data in real time, including air quality monitoring data and meteorological data, providing a comprehensive environmental information foundation for the system. Accurate predictions of air quality changes and pollution source distribution help governments formulate effective governance policies and reduce the impact of pollution on public health.

[0224] This invention compares the STFDG model with other deep learning models to verify the superiority of the proposed model in meteorological modeling applications. The comparison models include STGCN, ASTGCN, TGCN, DCRNN, and LSGCN. In the comparison experiments, all models used the same dataset and parameters, with a training-to-test set ratio of 8:2, an input length of 64, output lengths of 5, 10, and 15 respectively, a learning rate of 0.001, 100 iterations, a batch size of 144, 3 layers in all neural networks, and 96 hidden layer units. All deep learning comparison models underwent parameter optimization using the Adam optimization algorithm with an optimized learning rate of 0.001. All comparison experiments were independently repeated 10 times, and the average value was used as the final result.

[0225] Table 9 shows the prediction results of each model for dataset 3 at prediction step sizes of 5, 10, and 15. Specifically, at a step size of 5, STFDG reduced the MAE, RMSE, and SMAPE error metrics by 1.5%, 3.7%, and 2.5% respectively compared to the best-performing comparison model LSGCN. At a step size of 10, STFDG reduced the MAE, RMSE, and SMAPE metrics by 8.6%, 1.8%, and 6.6% respectively compared to the LSGCN model, and reduced the SMAPE metric by 5% compared to DCRNN. ​​At a step size of 15, STFDG reduced the MAE, RMSE, and SMAPE metrics by 21.3%, 36.8%, and 14.8% respectively compared to the best-performing comparison model LSGCN, reduced the RMSE by 34.9% compared to ASTGCN, and reduced the SMAPE by 8.4% compared to DCRNN.

[0226] Table 9. Data 3: Evaluation results of each model for different prediction lengths.

[0227]

[0228]

[0229] Figure 15 The bar charts show the evaluation indicators for different prediction step sizes. The comprehensive evaluation indicators and bar charts show that STFDG can provide more accurate modeling results in real air quality data, proving the effectiveness and generalization of the method.

[0230] To visually demonstrate the model's fitting performance, Figure 16 This is a comparison chart of the STFDG results and actual values ​​in an air quality data experiment. The chart shows the fitted curves for air quality, CO concentration, and PM2.5 concentration. The data shows the fitted results for different time lengths. It can be seen that the STFDG model can not only fit the changing trends of high-dimensional multi-factor data well, but also accurately follow the trend when the direction of data change suddenly changes. The performance is not significantly different on different datasets, further proving the model's generalization ability.

Claims

1. A multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network, characterized in that: First, missing data in the dataset is imputed using a low-rank tensor decomposition method based on the Schatten capped p-norm, leveraging the low-rank structure of the data. A threshold is introduced to adjust the norm's penalty for outliers, incorporating the Schatten capped p-norm. Second, by introducing a sparse error term E, the model can separate and process outliers from normal data. Third, an alternating direction multiplier parameter optimization method based on momentum-variance collaboration is constructed, utilizing momentum acceleration by introducing a momentum term and leveraging historical iteration information to accelerate the convergence of the current batch optimization algorithm. Finally, a graph neural network method based on spatiotemporal-frequency dynamic fusion is used to dynamically generate an adjacency matrix by learning node embeddings.

2. The multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network as described in claim 1, characterized in that, Includes the following steps: Step 1: Construct a low-rank tensor decomposition method based on the Schatten capped p-norm; Step 2: Construct a robust data missing imputation method for low-rank tensor decomposition; Step 3: Construct an alternating direction multiplier parameter optimization method based on momentum-variance coordination; Step 4: Temporal modeling based on spatial dynamic graph neural networks; Step 5: Frequency domain analysis and temporal convolution are the two core isomorphic modules; Step 6: A multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network.

3. The multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network as described in claim 2, characterized in that: Step 1 includes introducing the Schatten capped p-norm, which adjusts the penalty of the norm on singular values ​​by introducing a threshold. For tensors Schatten capped p-norm extension is: Where τ is a threshold used to filter out singular values, X (k) Let X be the matrix obtained by expanding tensor X along the k-th mode, where k is the weight parameter, satisfying... The objective function of the model is to minimize the Schatten capped p-norm of the tensor, while ensuring that the completed tensor is consistent with the observed partial data in terms of observation location. The mathematical expression is: Where X is a third-order tensor representing the complete data; Y is a partially observed tensor containing missing values; P Ω (·) is the orthogonal projection operator on the observation set Ω; It is the Schatten capped p-norm of the tensor; the Schatten capped p-norm of the tensor is defined as the weighted sum of the Schatten p-norms of each expanded matrix of the tensor, and the p-th power of the minimum value after comparing each singular value with the threshold τ. Where, σ i (X(k)) are the singular values ​​of the k-th expansion matrix of tensor X; τ is the threshold used to filter singular values; p∈(0,1] is a parameter that controls the strength of nonconvexity; α k It is the weight parameter of the k-th expanded matrix.

4. The multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network as described in claim 2, characterized in that: Step 2 includes proposing a low-rank tensor decomposition-Schatten capped p-norm data missing imputation method that introduces a sparse error term E for modeling and handling random outliers.

5. The multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network as described in claim 2, characterized in that: Step 3 includes variance reduction, which improves the convergence speed of the algorithm by reducing the variance of the gradient estimation, and is achieved by using SVRG gradient estimation.

6. The multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network as described in claim 2, characterized in that: Step 4 includes dynamically generating an adjacency matrix by learning node embeddings. Each node is assigned two embedding vectors, representing the features of the source node and the target node, respectively. For each time slice t, two vectors Θ and Ψ of length d are generated, where all elements are randomly initialized learnable parameters. Then, Θ is calculated. T Ψ serves as the initial adjacency matrix for a time slice, the values ​​of which are optimized through training.

7. The multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network as described in claim 2, characterized in that: Step 5 involves converting the one-dimensional sequence after FFT transformation into a set of two-dimensional tensors based on multiple periods, thereby extending the analysis of time-varying changes to a two-dimensional space. Through this transformation, the changes within the period are embedded into the columns of the two-dimensional tensor, and the changes between the periods are embedded into the rows. The two-dimensional change pattern of the sequence is modeled using a two-dimensional convolution kernel.

8. The multi-factor data prediction modeling method based on spatiotemporal frequency dynamic fusion graph neural network as described in claim 2, characterized in that: Step 6 includes concatenating the spatial features extracted by the spatial dynamic graph neural network and the temporal features extracted by the two core isomorphic modules of frequency domain analysis and temporal convolution, and outputting the result through a fully connected layer. In each layer, the output feature H of the dynamic graph isomorphic network... (l) The feature X output by the temporal convolution fast 2D convolution method (l+1) Interact to form a new feature representation Z (l+1) : Z (l+1) =Concat(H (l) ,X (l+1) ) In the final layer, the feature representations are further aggregated into a global feature representation through a global pooling operation: G (L) =GlobalPool(Z (L) ) Among them, Z (L) The fusion feature representation of the last layer of the formula, Finally, the global feature representation G (L) Passed to the output layer: Where FC represents a fully connected layer; b is a bias term.

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