A track tensor completion and anomaly repair method based on singular value weighted truncation

Abstract

The application discloses a track tensor completion and abnormality repairing method based on singular value weighting and truncation, and relates to the technical field of flight track data processing.The application fully utilizes easily-obtained ADS-B track data, and mines the internal law through intelligent learning capability; meanwhile, aiming at the complex problem that missing and abnormality are coupled with each other in the track data, a robust tensor model is innovatively constructed, which fuses an adaptive weight mechanism, singular value truncation and sparse abnormality constraint, so that the collaborative and accurate processing of missing completion and abnormality repairing is realized; further, an optimization solving strategy based on an alternating direction multiplier method is proposed, and through the augmented Lagrange method, low-rank track tensors, sparse abnormality tensors and auxiliary variables are efficiently block-iteratively optimized, so that the complete track can be accurately recovered in a complex data environment.

Description

Technical Field

[0001] This invention relates to the field of flight trajectory data processing technology, and in particular to a method for trajectory tensor completion and anomaly repair based on singular value weighted truncation. Background Technology

[0002] In the concept of next-generation air transport systems, Automatic Dependent Surveillance-Broadcast (ADS-B) is considered a primary air traffic surveillance technology. Aircraft equipped with ADS-B transmitters obtain their real-time flight positions via global navigation satellite systems and periodically broadcast this information to ground receiving stations and other aircraft equipped with ADS-B receivers. This enhances pilots' and air traffic controllers' awareness of the airspace situation, strengthens air-to-ground and air-to-air coordination, and improves air traffic efficiency. Compared to traditional radar surveillance technologies, ADS-B offers numerous advantages, including high positioning accuracy, rapid data updates, and low maintenance costs. With the continuous development and maturation of ADS-B technology, flight path data can be easily obtained through public channels such as Flight Aware and Flight Radar24. As a comprehensive record and objective description of aircraft flight processes, this flight path data can support research for various downstream applications, such as flight trajectory prediction, air traffic flow prediction, and flight conflict monitoring. In particular, with the rise of artificial intelligence and data mining technologies in the civil aviation field, flight path data used to train intelligent models has become especially important. However, due to limitations imposed by airborne equipment and internal system factors, as well as numerous external factors such as signal strength, communication links, and weather conditions, trajectory data inevitably exhibits skipped or missing points. These issues severely impact the reliability of trajectory data and have consistently been a major obstacle to its application. Therefore, it is necessary to combine trajectory completion models and anomaly repair models to improve the integrity and reliability of trajectory data.

[0003] In reality, the problems of missing and anomalies in flight track data are closely coupled and mutually influential. The accuracy of completing missing points is easily affected by anomalies in the flight track, while accurate repair of anomalies requires complete flight track data as a reference. However, most existing studies treat these two tasks separately. The representative track preprocessing method based on neural networks and Newton interpolation is one of the few techniques capable of simultaneously handling anomaly repair and missing point completion. Nevertheless, this method employs a sequential processing strategy of repairing anomalies first and then completing missing points. The interpolation accuracy of missing positions is easily affected by inaccurate identification of preceding anomalies, limiting its robustness in the face of the coupled problem of missing and anomaly data. Furthermore, the uneven distribution of flight track data across different flight phases (such as takeoff, landing, and cruise) further exacerbates this problem. When processing large-scale, frequently updated broadcast automatic dependent surveillance data, the excessively high time cost also affects the feasibility and scalability of the algorithm. Summary of the Invention

[0004] The purpose of this invention is to provide a method for track tensor completion and anomaly repair based on singular value weighted truncation, in order to improve the technical problems of existing track preprocessing methods when dealing with tightly coupled data loss and anomaly interference in ADS-B data. These methods are limited in completion accuracy and repair robustness due to the use of a serial separation strategy, and are unable to fully explore the low-rank structural features and deep correlation information of track data in multidimensional spatiotemporal dimensions.

[0005] To achieve the above-mentioned objectives, the embodiments of the present invention provide the following technical solutions:

[0006] A method for track tensor completion and anomaly repair based on singular value weighted truncation includes:

[0007] Flight trajectory data is collected and low-rank analysis is performed using the SVD decomposition method to generate low-rank decomposition results and trajectory tensor representations.

[0008] The trajectory tensor representation is decomposed and auxiliary variables are set. A low-rank regularization term is constructed by combining an adaptive weighting mechanism and a singular value truncation mechanism.

[0009] Based on auxiliary variables, sparse anomaly constraint terms, and low-rank regularization terms, a complete anomaly robust model is generated.

[0010] The robust model for completing anomalies is iteratively optimized using the alternating direction multiplier method to generate optimized trajectory tensors.

[0011] In the aforementioned process, this method introduces low-rank analysis based on singular value decomposition to accurately mine the deep correlation structure of track data in the spatiotemporal dimension, providing reliable priors for subsequent modeling. Based on this, a robust tensor model integrating adaptive weighting, singular value truncation, and sparse anomaly constraints is constructed. This model fully utilizes auxiliary variables to effectively separate low-rank and anomaly components, enabling collaborative handling of complex problems involving deep coupling between missing and anomaly components, avoiding the errors and insufficient robustness inherent in traditional sequential strategies. Simultaneously, the alternating direction multiplier method is employed for block-based iterative optimization, significantly improving computational efficiency and scalability while ensuring solution accuracy. Therefore, this method can accurately recover the complete track structure in operating environments where ADS-B track data exhibits widespread missing and anomaly interference, effectively improving data quality and usability. It provides high-quality data support for high-level intelligent applications such as flight trajectory prediction and air traffic flow analysis, effectively solving the technical challenges of limited accuracy, poor robustness, and insufficient multi-dimensional feature mining in existing track preprocessing methods when dealing with missing and anomaly coupling problems.

[0012] Furthermore, the construction of the low-rank regularization term includes:

[0013] The track tensor representation is decomposed to generate a low-rank track tensor and a sparse anomaly tensor.

[0014] Auxiliary variables are generated based on low-rank track tensors and sparse anomaly tensors, combined with the first and second auxiliary constraints.

[0015] A low-rank regularization term is constructed based on an adaptive weighting mechanism and a singular value truncation mechanism.

[0016] Furthermore, the formula corresponding to the low-rank regularization term is:

[0017] ;

[0018] in, Indicates along the first Low-rank trajectory tensor The result of the expansion is the first matrix, Indicates the adjustment parameter. This represents the rank of the track matrix representation. This represents the singular value truncation threshold. Represents a low-rank regular term. This represents the summation function. Indicates the first A singular value, Indicates the position number of a single flight path.

[0019] In the aforementioned process, this method constructs a low-rank regularization term that integrates an adaptive weighting mechanism and a singular value truncation mechanism, thereby achieving a refined characterization and differentiated constraints on the multidimensional structural features of flight track data. By introducing adjustment parameters and a singular value truncation threshold, this method can dynamically adjust the regularization strength based on the actual distribution characteristics of each modality's data, while effectively truncating extremely small singular values. This allows for the precise removal of noise and redundant components while preserving the main spatiotemporal structure of the flight track, solving the technical challenge in traditional low-rank constraints where "homogenization penalties lead to excessive weakening of primary features and difficulty in effectively suppressing secondary interference." This significantly improves the ability of the robust model for completing anomalies to uncover deep correlations in flight track data and enhances its robustness against interference.

[0020] Furthermore, the generation of the robust anomaly model includes:

[0021] Set an adaptive threshold, and according to the formula:

[0022] ;

[0023] Construct sparse anomaly constraints; where, Represents the quantile parameter. This represents a sparse anomaly constraint term. Represents absolute value. Indicates an adaptive threshold. This represents the summation function. Indicates the first A sparse anomaly tensor Indicates the first A sparse anomalous tensor exist The element value at that position;

[0024] Based on auxiliary variables, sparse anomaly constraints, and low-rank regularization terms, a complete anomaly robust model is generated.

[0025] Furthermore, the generated trajectory tensor optimization result includes:

[0026] Based on the first auxiliary constraint, the augmented Lagrangian method is used to process the complete anomaly robust model to generate the objective function;

[0027] The objective function is decomposed to generate the first iteration problem, the second iteration problem, the third iteration problem, and the fourth iteration problem;

[0028] Initialize the low-rank track tensor, sparse anomaly tensor, auxiliary variables, and Lagrange multipliers;

[0029] The first, second, third, and fourth iteration problems are solved iteratively using the alternating direction multiplier method to generate trajectory tensor optimization results.

[0030] Furthermore, the iterative solution of the first, second, third, and fourth iteration problems using the alternating direction multiplier method includes:

[0031] Update the penalty factors for the current iteration; normalize the aircraft position information in the low-rank track tensor and sparse anomaly tensor to generate the corresponding normalized position data, and update the low-rank track tensor and sparse anomaly tensor.

[0032] Based on the problem of the first iteration, update the low-rank trajectory tensor under different modes in the current iteration;

[0033] Based on the second iteration problem and each updated low-rank track tensor, update the auxiliary variables for the current iteration;

[0034] Based on the third iteration problem and the updated auxiliary variables, update the sparse anomaly tensors in different modalities of the current iteration;

[0035] Based on the fourth iteration problem and each updated sparse anomaly tensor, update the Lagrange multipliers for different modes in the current iteration;

[0036] Based on the updated low-rank track tensors, updated auxiliary variables, updated sparse outlier tensors, and updated Lagrange multipliers, the normalized position data are denormalized and it is determined whether the iteration conditions are met, generating track tensor optimization results.

[0037] Furthermore, the formula for updating the low-rank track tensor under different modes in the current iteration is:

[0038] ;

[0039] in, This represents the updated low-rank track tensor. , These represent the update matrices respectively. The left singular vector matrix and the right singular vector matrix, Represents the optimal set of singular values. Represents the transpose matrix. This represents the tensor folding operator. This represents a diagonal matrix.

[0040] Furthermore, the formula corresponding to the updated auxiliary variable for the current iteration is:

[0041] ;

[0042] in, This represents the updated auxiliary variable. This represents the summation function. Indicates the first Penalty factor under each modality Indicates the first The iteration corresponding to the first A modal Lagrange multiplier, This represents the updated low-rank track tensor. Indicates the first The sparse anomaly tensor corresponding to the next iteration.

[0043] In the above process, this method introduces an adaptive weighting mechanism to dynamically adjust the regularization intensity based on the differences in the contribution of different modes to low-rank properties, thereby more accurately preserving the main spatiotemporal structural features of the track. A singular value truncation mechanism is employed to directly filter out small singular values ​​reflecting noise or minor details, focusing on the parts that significantly contribute to the main structure, thus significantly improving the completion accuracy. A sparse anomaly constraint term is constructed, combined with quantile parameters and adaptive thresholds, to accurately identify and separate discrete anomalies while protecting normal track data from interference. The introduction of auxiliary variables and the use of alternating direction multipliers for block-based iterative optimization effectively decouples the complex coupling relationships between multimodal expansion matrices, enabling the model to achieve efficient solutions while ensuring data consistency. Furthermore, the augmented Lagrange method is used to collaboratively iteratively update the low-rank track tensor, sparse anomaly tensor, and Lagrange multipliers, further enhancing the model's robustness and convergence stability in complex data environments.

[0044] In summary, this method not only solves the problems of difficulty in handling missing and anomaly coupling and insufficient mining of multidimensional low-rank structures in existing technologies, but also significantly improves the accuracy of trajectory data completion and anomaly repair capability through the synergistic effect of various feature modules. It provides a high-quality data foundation for downstream applications such as flight trajectory prediction and air traffic flow analysis, and strongly supports the reliable operation and development of intelligent aviation systems. Attached Figure Description

[0045] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0046] Figure 1 This is a flowchart of the method in an embodiment of the present invention. Detailed Implementation

[0047] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0048] Please see Figure 1 This embodiment provides a method for track tensor completion and anomaly repair based on singular value weighted truncation. Figure 1 The execution entity of the method shown can be a software and / or hardware device. The execution entity of this application can include, but is not limited to, at least one of the following: user equipment, network equipment, etc. User equipment can include, but is not limited to, computers, smartphones, personal digital assistants (PDAs), and the aforementioned electronic devices. Network equipment can include, but is not limited to, a single network server, a server group consisting of multiple network servers, or a cloud based on cloud computing consisting of a large number of computers or network servers. Cloud computing is a type of distributed computing, consisting of a super virtual computer composed of a group of loosely coupled computers. This embodiment does not impose any limitations on this.

[0049] A method for track tensor completion and anomaly repair based on singular value weighted truncation includes:

[0050] S1. Collect flight trajectory data and perform low-rank analysis using the SVD decomposition method to generate low-rank decomposition results and trajectory tensor representation;

[0051] Specifically, flight trajectory data for the same flight number is collected from the Automatic Dependent Surveillance-Broadcast (ADS-B) system, and the flight trajectory data is constructed into a trajectory matrix representation.

[0052] To verify that the track matrix representation has a good low-rank structure, the track matrix representation is decomposed using the Singular Value Decomposition (SVD) method. For a track matrix containing... strip Track matrix representation of dimensional track data According to the formula:

[0053] ;

[0054] Representation of the track matrix Decompose it. Among them, , They represent the first orthogonal singular vectors ( ) and the second orthogonal singular vector ( ), Represents a positive real diagonal matrix ( ), Represents the transpose matrix. Represents a constant. Representation of the track matrix Rank.

[0055] Positive real diagonal matrix The diagonal elements are singular values. They are usually arranged in descending order, that is:

[0056] ;

[0057] Represents a diagonal matrix. , , They represent the first singular value, the second singular value, and the third singular value, respectively. A singular value. ,when much smaller and When this happens, the track matrix is ​​represented. As a low-rank matrix. Furthermore, if a matrix is ​​a low-rank matrix, its front... The energy of each singular value is close to the total energy, that is... . This represents the summation function. Indicates the first A singular value.

[0058] Therefore, it can be seen that the matrix representation corresponding to the flight trajectory data in this embodiment has low rank, and before use... The ratio of the energy of a few singular values ​​to the total energy is used to verify the low-rank nature of the track matrix representation. SVD decomposition of the track matrix representation reveals that its energy is mainly concentrated in a few singular values, indicating that the track set exhibits significant low-rank characteristics in its structure. This low-rank characteristic is used as the result of low-rank decomposition, thus providing a priori evidence for subsequent track tensor modeling, track data completion, and anomaly repair.

[0059] Then, according to the formula:

[0060] ;

[0061] The flight trajectory data is converted into a top-down hierarchical structure. The set of tracks described. Among them, , , , These represent the first flight, the second flight, and the third flight, respectively. The next flight and the Flight trajectory data corresponding to this flight , , , Representing the positions of the first aircraft, the second aircraft, and the third aircraft respectively. The aircraft's position and the first aircraft position points , , These represent the longitude, latitude, and altitude of a specific aircraft location. Each flight trajectory data point encompasses all aircraft location information related to the entire flight time, from the departure airport to the destination airport. The corresponding aircraft position information is obtained from each equally spaced aircraft.

[0062] Low-rank decomposition results show that the tracks of the same flight number follow similar spatiotemporal patterns on different dates, and each location point is highly correlated with its neighbors in the spatiotemporal dimension. Based on the low-rank decomposition results, the track set is further modeled as a three-dimensional tensor. This refers to the track tensor representation. This represents the dimension of a sequence of flight paths with the same flight number on different dates. This represents the attribute feature dimension of each location point. This represents the position sequence dimension of a single flight path. It represents the set of real numbers. Attribute features include the longitude, latitude, and altitude information of the location point.

[0063] Track tensor representation It can fully characterize the high-order correlation structure of multimodal and multi-mode data of flight paths, thereby more accurately reflecting the global spatiotemporal characteristics of flight paths.

[0064] S2. Decompose the track tensor representation and set auxiliary variables. Combine the adaptive weighting mechanism and the singular value truncation mechanism to construct a low-rank regularization term.

[0065] S2 includes:

[0066] S2-1. Decompose the track tensor representation to generate a low-rank track tensor and a sparse anomaly tensor.

[0067] Low-rank track tensors are used to characterize the main spatiotemporal structural features of flight trajectory data; sparse anomaly tensors are used to characterize a small number of outliers or error components in the track, so as to effectively distinguish between normal track information and abnormal interference.

[0068] S2-2. Based on the low-rank track tensor and the sparse anomaly tensor, and combined with the first and second auxiliary constraints, auxiliary variables are generated.

[0069] It needs to be explained that, due to the low-rank track tensor The expansion matrix along the three modes ( , , Because they share tensor elements and thus have a certain coupling relationship, if the low-rank track tensor and the sparse anomaly tensor are directly used to construct a robust anomaly model, the three expanded matrices must simultaneously satisfy the low-rank constraint and the data consistency constraint, forming a complex coupled optimization problem that is difficult to solve in a closed form. Therefore, this embodiment introduces an auxiliary variable, which acts as an intermediary carrier of observation information. The first auxiliary constraint ensures the consistency of the observation data. Then, the second auxiliary constraint is used to ensure that the three low-rank tensors ( , , Each of them can be independently optimized in its corresponding expansion matrix. , , This transforms the three originally coupled expansion matrices into independently solvable low-rank optimization subproblems, thereby achieving decoupling and completion of each expansion matrix. Simultaneously, the second auxiliary constraint decomposes the observation information in the flight trajectory data into low-rank and anomalous components, thus enabling the separation and repair of anomalous components.

[0070] The formula corresponding to the first auxiliary constraint is:

[0071] ;

[0072] The formula corresponding to the second auxiliary constraint is:

[0073] ;

[0074] in, Represents auxiliary variables. This represents the set of observed trajectory locations. This indicates that the flight trajectory data contains several missing location points and anomalous disturbance points in the partial observation tensor (the trajectory tensor to be completed and anomaly repaired), and , Represents the set of points on the trajectory. orthogonal projection, Indicates the first A low-rank track tensor Indicates the first A sparse anomalous tensor.

[0075] Orthographic projection The corresponding formula is:

[0076] ;

[0077] in, Tensor exist The element value at that position, Indicates the position number of a single flight path.

[0078] S2-3. Based on the adaptive weighting mechanism and the singular value truncation mechanism, construct a low-rank regularization term.

[0079] It should be explained that, since the results obtained through singular value decomposition show that the energy of the flight track data is highly concentrated in the first few singular values, exhibiting significant low-rank characteristics, this means that the main structure of the track can be characterized by a small number of key components. Traditional low-rank constraints (such as the nuclear norm) uniformly shrink all singular values, which can easily overly weaken large singular values ​​representing the main spatiotemporal features while suppressing noise, and also cannot effectively remove redundant small singular values. Therefore, this embodiment introduces a truncation mechanism, which aims to directly ignore extremely small singular values ​​that reflect noise or minor details, focusing the constraints on the parts that contribute significantly to the main structure.

[0080] Meanwhile, the track tensor has three physically distinct dimensions (number of flights, spatial coordinates, and time series), each contributing differently to low-rank performance. Applying the same regularization weights would distort the inherent multidimensional relationships in the data. Therefore, this embodiment also introduces an adaptive weighting mechanism, enabling the model to dynamically adjust the low-rank penalty intensity based on the actual distribution characteristics of each modality during optimization. Combining the adaptive weighting mechanism and the singular value truncation mechanism, a low-rank regularization term can be constructed that can selectively filter out redundant interference and differentiate dimensional importance for structured constraints, thereby significantly improving the accuracy and robustness of track completion and anomaly repair.

[0081] The low-rank regularization term measures low-rank property by the weighted sum of the singular values ​​of the expanded matrix, and the corresponding formula is:

[0082] ;

[0083] in, Indicates along the first Low-rank trajectory tensor The result of the expansion is the first matrix, Indicates the adjustment parameter. This represents the singular value truncation threshold. This indicates a low-rank regular term.

[0084] Adjust parameters The value range is [0, 1], and it is used to dynamically adjust the rank approximation method to depict the low-rank structure and spatiotemporal characteristics of the track data. Singular value truncation threshold. It is used to truncate smaller singular values ​​to reduce the influence of noise and minor components, thereby improving the accuracy and robustness of track completion.

[0085] S3. Generate a complete anomaly robust model based on auxiliary variables, sparse anomaly constraint terms, and low-rank regularization terms.

[0086] S3 includes:

[0087] S3-1. Set an adaptive threshold and construct sparse anomaly constraint terms;

[0088] The formula corresponding to the sparse anomaly constraint term is:

[0089] ;

[0090] in, This represents the quantile parameter, which ranges from (0, 1) and is used to control the intensity of differentiated penalties for positive and negative anomalies. Indicates sparse anomaly constraints; Represents absolute value; This represents an adaptive threshold used to distinguish between small and large anomalies, and it automatically adjusts based on the data distribution of the anomaly tensor. Its expression is: , Represents the median function; Indicates the first A sparse anomalous tensor exist The element value at that location.

[0091] This embodiment introduces a sparse anomaly constraint term. Through the synergistic effect of quantile parameters and adaptive thresholds, it addresses the sparse anomaly tensor. Differential penalties are imposed on the elements in the sparse anomaly tensor, meaning that small fluctuations are completely suppressed (set to zero) within a threshold, while large anomalies undergo nonlinear contraction based on the degree and direction of deviation. This mechanism forces the sparse anomaly tensor to... It exhibits non-zero values ​​only at a very few locations, thus enabling accurate identification and effective suppression of discrete anomalies while ensuring the integrity of the main track structure.

[0092] S3-2. Based on auxiliary variables, sparse anomaly constraint terms, and low-rank regularization terms, generate a complete anomaly robust model. The corresponding formula is:

[0093] ;

[0094] in, Describes the minimum value function; Indicates along the first The expansion matrix of each modality expansion Weight parameters ( )and ; Represents the weighting coefficients used to adjust the sparse outlier tensor. The constraint strength in the complete anomaly robust model is increased so that the model can maintain the main variation law of the trajectory and effectively suppress the influence of anomalies.

[0095] S4. The robust model for anomaly completion is iteratively optimized using the alternating direction multiplier method to generate the optimized track tensor. The optimized track tensor is the complete flight track data after iterative completion and anomaly repair.

[0096] S4 includes:

[0097] S4-1. Based on the first auxiliary constraint, the augmented Lagrangian method is used to process the complete anomaly robust model to generate the objective function;

[0098] The formula corresponding to the objective function is:

[0099] ;

[0100] in, Indicates the first Penalty factor under each modality Indicates the first Lagrange multipliers in each modality (used to dynamically balance the differences between dimensions). The square of the Frobenius norm of the inner product of two tensors is given. Describe the objective function. This represents the inner product of two tensors.

[0101] S4-2. Decompose the objective function to generate the first iteration problem, the second iteration problem, the third iteration problem, and the fourth iteration problem;

[0102] The formula corresponding to S4-2 is:

[0103] ;

[0104] in, , Let represent the sparse anomaly tensor and the low-rank track tensor, respectively. , They represent the first The second iteration and the first The iteration corresponding to the first A sparse anomaly tensor of each modality This represents a function that achieves its minimum value. , They represent the first The second iteration and the first The iteration corresponding to the first A modal Lagrange multiplier, , They represent the first The second iteration and the first The auxiliary variables corresponding to the next iteration Indicates the first The iteration corresponding to the first The low-rank trajectory tensor of each mode. The sub-formulas from top to bottom in this formula represent the first iteration problem, the second iteration problem, the third iteration problem, and the fourth iteration problem, respectively.

[0105] S4-3. Initialize the low-rank track tensor, sparse anomaly tensor, auxiliary variables, and Lagrange multipliers;

[0106] S4-4. The first, second, third, and fourth iteration problems are solved iteratively using the alternating direction multiplier method to generate the trajectory tensor optimization results.

[0107] S4-4 includes:

[0108] S4-4-1. According to the formula:

[0109] ;

[0110] Update the penalty factors for the current iteration; where, , They represent the first The second iteration and the first The iteration corresponding to the first One penalty factor; This represents the upper bound of the preset penalty factor, used to limit the growth rate of the penalty factor during the iteration process to ensure the stability of the algorithm iteration; its value can be set by the user according to the data scale and convergence speed requirements.

[0111] S4-4-2. Normalize the aircraft position information in the low-rank track tensor and sparse anomaly tensor to generate the corresponding normalized position data and update it to the low-rank track tensor and sparse anomaly tensor.

[0112] Since the longitude, latitude, and altitude values ​​of the track positions vary, to eliminate the influence of dimensional differences on subsequent track tensor modeling and iterative solution, the longitude, latitude, and altitude in each low-rank track tensor or sparse outlier tensor are normalized using Min-Max. Therefore, the formula corresponding to S4-4-2 is:

[0113] ;

[0114] in, , , They represent the first The normalized longitude, latitude, and altitude corresponding to each aircraft location point , , They represent the first The longitude, latitude, and altitude corresponding to each aircraft's location point. , These represent the minimum and maximum longitude values, respectively. , These represent the minimum and maximum latitude values, respectively. , These represent the minimum and maximum height values, respectively.

[0115] The normalized position data is updated to the low-rank track tensor and the sparse anomaly tensor.

[0116] S4-4-3. Based on the problem of the first iteration, update the low-rank trajectory tensor under different modes in the current iteration;

[0117] It needs to be explained that, with the first Taking the first mode as an example, in each iteration, the update of the low-rank track tensor can be regarded as an optimization problem with adaptive weighted singular value truncation regularization, so as to make the rank of the expansion matrix of the first mode as low as possible while maintaining consistency with the observed track data.

[0118] The first iteration problem The expansion is:

[0119] ;

[0120] in, This represents the preset regularization parameter, and its expression is: This is used to balance the fit between the adaptive weighted singular value truncation regularization term and the track data. Its value can be set by the user according to the data scale and recovery accuracy requirements.

[0121] To facilitate the solution, in the first... Expanding the low-rank track tensor in several dimensions, i.e., updating the matrix. The value assigned is the first matrix And the original matrix .in, Represents the original matrix, Indicates the first The first mode A Lagrange multiplier, Indicates the first The first mode A sparse anomaly tensor Indicates the first The first mode Auxiliary variables.

[0122] Original matrix It uses the alternating direction multiplier method to update the low-rank track tensor Intermediate variables introduced during the process. Original matrix. It consists of quantities determined in the previous iteration, representing the initial state before optimization of the current subproblem. Introducing the original matrix... The goal is to transform the tensor optimization problem into a matrix optimization form, which facilitates SVD decomposition and the use of an adaptive weighted singular value truncation strategy to solve for the optimal low-rank matrix.

[0123] The expansion of the first iteration problem is then equivalently transformed into a matrix optimization form, with the corresponding formula being:

[0124] ;

[0125] For the original matrix Perform SVD decomposition to obtain the original decomposition result, i.e.:

[0126] ;

[0127] in, , Representing the original matrix respectively The left singular vector matrix and the right singular vector matrix.

[0128] Due to the update matrix With the original matrix If the singular value vectors have the same direction, they can be decomposed into:

[0129] ;

[0130] in, , These represent the update matrices respectively. The left singular vector matrix and the right singular vector matrix, , , These represent the singular values ​​after the first update, the singular values ​​after the second update, and the singular values ​​after the third update, respectively. The updated singular value.

[0131] For each singular value By solving independently and generating the corresponding optimal solutions, the optimal set of singular values ​​can be obtained. To form the optimal singular value vector .

[0132] Based on the obtained optimal singular value vector According to the formula:

[0133] ;

[0134] Update the current iteration's... The low-rank track tensor for each modality is used to obtain the corresponding updated low-rank track tensor. .in, Represents a tensor folding operator, used along the folding line. The dimension transforms the matrix into a third-order tensor.

[0135] Other modalities involve updating the low-rank trajectory tensor in the same way as updating the first... The low-rank track tensor is expressed in exactly the same way across all modes.

[0136] S4-4-4. Based on the second iteration problem and each updated low-rank track tensor, update the auxiliary variables for the current iteration;

[0137] It needs to be explained that by fixing other variables, the optimization problem of the second iteration can be transformed into a weighted least squares problem with a closed-form solution, according to the formula:

[0138] ;

[0139] Update the auxiliary variable in the current iteration to obtain the updated auxiliary variable. .

[0140] S4-4-5. Based on the third iteration problem and the updated auxiliary variables, update the sparse anomaly tensors in different modalities of the current iteration;

[0141] It needs to be explained that, with the first Taking a single modality as an example, in each iteration, updating the sparse anomaly tensor can be viewed as an optimization problem that achieves sparse anomaly constraints through quantile weighting and adaptive thresholding. This aims to maintain consistency with the observed track data while ensuring the sparse anomaly tensor... Non-zero values ​​are taken only at a few locations to achieve effective separation and suppression of anomalous disturbances.

[0142] The third iteration problem The expansion is:

[0143] ;

[0144] in, Quantile parameter For sparse anomalous tensors The applied sparse anomaly constraint.

[0145] To facilitate the solution, let the intermediate tensor be... Equivalent to Therefore, the expansion of the third iteration problem is equivalently transformed into:

[0146] ;

[0147] The formula after the third iteration problem is transformed consists of two parts: one is used to make the sparse outlier tensor... As close as possible to the intermediate tensor Secondly, it is used to make sparse anomalous tensors It takes non-zero values ​​only at a small number of locations. Therefore, it can be decomposed into an independent one-dimensional minimization problem that is solved separately for each location, thus obtaining the sparse outlier tensor. The closed-loop update rule. Let the positive threshold be... for negative threshold for Then for the intermediate tensor Each element is shrunk and updated according to a threshold: when the element value is within the range When the value is outside the range, the corresponding outlier is updated to 0; when the element value exceeds the range, it is updated according to the quantile parameter. The settings allow for differentiated shrinkage, retaining only the portion exceeding the threshold.

[0148] Therefore, according to the formula:

[0149] ;

[0150] Update the current iteration's... The sparse anomaly tensor under each modality is used to obtain the corresponding updated sparse anomaly tensor. .in, This represents the Hadamard product (i.e., element-wise multiplication of tensors of the same dimension). Represents the maximum value function. The sign function is used to extract the sign information of elements in the expression, and its expression is:

[0151] .

[0152] Other modalities involve updating sparse outlier tensors in the same way as updating the first tensor. The way sparse anomaly tensors are expressed is exactly the same across all modalities.

[0153] S4-4-6. Based on the fourth iteration problem and each updated sparse anomaly tensor, update the Lagrange multipliers for different modes in the current iteration;

[0154] It needs to be explained that, with the first Taking a modality as an example, the update of the Lagrange multipliers is used to apply consistency constraints. The deviations are cumulatively corrected, thereby gradually increasing the degree to which the constraints are satisfied. Then, according to the formula...

[0155] ;

[0156] Update the current iteration's... The Lagrange multipliers for each modality are used to obtain the corresponding updated Lagrange multipliers. .

[0157] The method of updating Lagrange multipliers in other modalities is the same as updating the first... The Lagrange multipliers in each modality behave exactly the same.

[0158] S4-4-7. Based on each updated low-rank track tensor, updated auxiliary variables, updated sparse outlier tensors, and updated Lagrange multipliers, denormalize each normalized position data and determine whether the iteration conditions are met, generating track tensor optimization results.

[0159] Specifically, since longitude, latitude, and altitude have been Min-Max normalized, in order to restore the iteration results (each updated low-rank track tensor and each updated sparse anomaly tensor) to their original dimensions and value range and to facilitate comparison and error assessment with the original track data, it is necessary to perform inverse normalization on the iteration results. That is, we can reverse the normalization formula in S4-4-2 to obtain each inverse normalized low-rank track tensor and each inverse normalized sparse anomaly tensor.

[0160] Determine whether the iteration result under the current iteration meets the convergence condition; if so, take the iteration result under the current iteration as the iteration optimization result (track tensor optimization result); otherwise, increment the current iteration count by 1 and return to S4-4-1 to iterate again.

[0161] The formula corresponding to the convergence condition is:

[0162] ;

[0163] in, This indicates the preset maximum number of iterations, used to limit the computational cost and running time of the iteration process; This represents the preset error threshold, used to control the accuracy of iteration termination. The values ​​of both can be set by the user based on the data size and convergence accuracy requirements. Relative error is used to measure the magnitude of change between two adjacent iterations to determine whether the iteration process tends to stabilize. The corresponding formula is:

[0164] ;

[0165] in, This represents the optimal tensor. The optimal tensor is the low-rank track tensor before the current iteration and before it was updated.

[0166] In summary, this invention fully utilizes readily available ADS-B track data and leverages intelligent learning capabilities to uncover its inherent patterns. Simultaneously, addressing the complex issue of coupled missing and anomaly data in track data, it innovatively constructs a robust tensor model integrating adaptive weighting, singular value truncation, and sparse anomaly constraints, achieving coordinated and precise processing of missing data completion and anomaly repair. Furthermore, it proposes an optimization strategy based on the alternating direction multiplier method, employing the augmented Lagrangian method to efficiently perform block-based iterative optimization of the low-rank track tensor, sparse anomaly tensor, and auxiliary variables, thereby accurately recovering complete tracks in complex data environments. Ultimately, this effectively improves the quality and usability of track data, laying a crucial foundation for enhancing the reliability and intelligence of downstream applications.

[0167] It should be noted that the specific methods by which each module performs operations in the system described in the above embodiments have been described in detail in the embodiments related to the method, and will not be elaborated here.

[0168] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

[0169] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A track tensor completion and anomaly repair method based on singular value weighted truncation, characterized in that, include: Flight trajectory data is collected and low-rank analysis is performed using the SVD decomposition method to generate low-rank decomposition results and trajectory tensor representations. The trajectory tensor representation is decomposed and auxiliary variables are set. A low-rank regularization term is constructed by combining an adaptive weighting mechanism and a singular value truncation mechanism. The construction of the low-rank regularization term includes: S2-1. Decompose the track tensor representation to generate a low-rank track tensor and a sparse anomaly tensor. S2-2. Based on the low-rank track tensor and the sparse anomaly tensor, and combined with the first and second auxiliary constraints, auxiliary variables are generated. The formula corresponding to the first auxiliary constraint is: ； The formula corresponding to the second auxiliary constraint is: ； in, Represents auxiliary variables. This represents the set of observed trajectory locations. This indicates that the flight trajectory data contains several missing location points and anomalous disturbance points in the partial observation tensor (the trajectory tensor to be completed and anomaly repaired), and , Represents the set of points on the trajectory. orthogonal projection, Indicates the first A low-rank track tensor Indicates the first A sparse anomalous tensor; S2-3. Construct a low-rank regularization term based on the adaptive weighting mechanism and the singular value truncation mechanism; The formula corresponding to the low-rank regularization term is: ； in, Indicates along the first Low-rank trajectory tensor The result of the expansion is the first 1 matrix Indicates the adjustment parameter. This represents the rank of the track matrix representation. This represents the singular value truncation threshold. Represents a low-rank regular term. This represents the summation function. Indicates the first A singular value, Indicates the position number of a single flight path, and adjusts the parameters. The value range is [0, 1]; Based on auxiliary variables, sparse anomaly constraint terms, and low-rank regularization terms, a complete anomaly robust model is generated. The generated and completed robust anomaly model includes: S3-1. Set an adaptive threshold and construct sparse anomaly constraint terms; The formula corresponding to the sparse anomaly constraint term is: ； in, This represents the quantile parameter, which ranges from (0, 1) and is used to control the intensity of differentiated penalties for positive and negative anomalies. Indicates sparse anomaly constraints; Represents absolute value; This represents an adaptive threshold used to distinguish between small and large anomalies, and it automatically adjusts based on the data distribution of the anomaly tensor. Its expression is: , Represents the median function; Indicates the first A sparse anomalous tensor exist The element value at that position; S3-2. Based on auxiliary variables, sparse anomaly constraint terms, and low-rank regularization terms, generate a complete anomaly robust model. The corresponding formula is: ； in, Describes the minimum value function; Indicates along the first The expansion matrix of each modality expansion Weight parameters ( )and ; Represents the weighting coefficients used to adjust the sparse outlier tensor. The constraint strength in the complete anomaly robust model is increased so that the model can maintain the main variation law of the trajectory and effectively suppress the influence of anomalies. The robust model for completing anomalies is iteratively optimized using the alternating direction multiplier method to generate optimized trajectory tensors.

2. The method for track tensor completion and anomaly repair based on singular value weighted truncation according to claim 1, characterized in that, The optimized results of the generated trajectory tensor include: Based on the first auxiliary constraint, the augmented Lagrangian method is used to process the complete anomaly robust model to generate the objective function; The objective function is decomposed to generate the first iteration problem, the second iteration problem, the third iteration problem, and the fourth iteration problem; Initialize the low-rank track tensor, sparse anomaly tensor, auxiliary variables, and Lagrange multipliers; The first, second, third, and fourth iteration problems are solved iteratively using the alternating direction multiplier method to generate trajectory tensor optimization results.

3. The method for track tensor completion and anomaly repair based on singular value weighted truncation according to claim 2, characterized in that, The iterative solution of the first, second, third, and fourth iteration problems using the alternating direction multiplier method includes: Update the penalty factors for the current iteration; normalize the aircraft position information in the low-rank track tensor and sparse anomaly tensor to generate the corresponding normalized position data, and update the low-rank track tensor and sparse anomaly tensor. Based on the problem of the first iteration, update the low-rank trajectory tensor under different modes in the current iteration; Based on the second iteration problem and each updated low-rank track tensor, update the auxiliary variables for the current iteration; Based on the third iteration problem and the updated auxiliary variables, update the sparse anomaly tensors in different modalities of the current iteration; Based on the fourth iteration problem and each updated sparse anomaly tensor, update the Lagrange multipliers for different modes in the current iteration; Based on the updated low-rank track tensors, updated auxiliary variables, updated sparse outlier tensors, and updated Lagrange multipliers, the normalized position data are denormalized and it is determined whether the iteration conditions are met, generating track tensor optimization results.

4. The method for track tensor completion and anomaly repair based on singular value weighted truncation according to claim 3, characterized in that, The formula for updating the low-rank track tensor under different modes in the current iteration is: ； in, This represents the updated low-rank track tensor. , These represent the update matrices respectively. The left singular vector matrix and the right singular vector matrix, Represents the optimal set of singular values. Represents the transpose matrix. This represents the tensor folding operator. This represents a diagonal matrix.

5. The method for track tensor completion and anomaly repair based on singular value weighted truncation according to claim 3, characterized in that, The formula for updating the auxiliary variable in the current iteration is: ； in, This represents the updated auxiliary variable. This represents the summation function. Indicates the first Penalty factor under each modality Indicates the first The iteration corresponding to the first A modal Lagrange multiplier, This represents the updated low-rank track tensor. Indicates the first The sparse anomaly tensor corresponding to the next iteration.

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