Precision monitoring method and system for air pollutant concentration based on tensor representation

Abstract

The present invention belongs to the technical field of air pollution monitoring, and specifically relates to a method and system for accurately monitoring air pollutant concentrations based on tensor representation. The method comprises: acquiring air quality data of a point to be detected in real time, inputting the acquired air quality data into a monitoring model to obtain monitoring results; marking the monitoring point according to the monitoring results; the present invention performs tensor completion by fusing multi-source data and spatiotemporal air pollutant concentration data to obtain high-precision tensor data.

Description

Technical Field

[0001] The present invention belongs to the technical field of air pollution monitoring, and specifically relates to a method and system for accurately monitoring air pollutant concentration based on tensor representation. Background Art

[0002] With the progress of urbanization and industrialization, more and more environmental pollution problems have also attracted public attention. Air pollution is an important source of environmental pollution that affects residents' health. In order to monitor and prevent air pollutants, many cities have established their own air quality monitoring stations, which will obtain the concentration of air pollutants in the city in real time. By analyzing and studying the concentration of air pollutants in the city, scientific research institutions can effectively assist the government in formulating environmental protection policies that are in line with the public interest.

[0003] Grid monitoring requires the acquisition of complete spatial data of the monitoring area. Since air quality monitoring stations can only cover a certain monitoring area, sparse air quality monitoring stations cannot completely cover a city. Therefore, various spatial interpolation algorithms are used to generalize sparse station monitoring data to the entire city plane. This process is called spatial resolution improvement. These algorithms are mainly divided into two categories. One is statistical algorithms, including Kriging interpolation and Inverse Distance Weighted; the other is machine learning algorithms, including Random Forest, Multilayer Perceptron and Neural Network. The correlation between distance and air pollutant concentration is a basic assumption of statistical algorithms. This simple assumption cannot well reflect the mechanism of spatial distribution of air pollutants, and often cannot consider the time-varying characteristics of air pollutant concentrations. Machine learning algorithms can combine the historical concentrations of air pollutants and fuse multiple air pollutant concentrations to improve the spatial resolution of air pollutant concentrations. However, since machine learning algorithms focus on fitting data and their non-intuitive internal mechanisms, the computational complexity of machine learning models is relatively high. In addition, machine learning algorithms often simply use historical data on air pollutants as input to the model, and the time-varying patterns of air pollutants cannot be directly described. This makes improving the spatial resolution of air pollutant concentrations and analyzing the time-varying characteristics of air pollutant concentrations two independent issues. Summary of the invention

[0004] In order to solve the problems existing in the above prior art, the present invention proposes a method for accurately monitoring the concentration of air pollutants based on tensor representation, which comprises: acquiring air quality data of a point to be detected in real time, inputting the acquired air quality data into a monitoring model to obtain a monitoring result; marking the monitoring point according to the monitoring result;

[0005] The process of processing air quality data using the monitoring model includes:

[0006] S1: The air quality data is represented by tensor; the tensor representation of the monitoring point includes the spatial distribution of the station, the time series of pollutant concentration, and the pollutant concentration;

[0007] S2: Construct a tensor completion optimization problem based on the tensor representation of air quality;

[0008] S3: According to the tensor completion optimization problem, the tensor singular value decomposition iterative update algorithm is used to perform singular value decomposition on the tensor representation data to obtain spatial singular functions, temporal singular functions and singular vectors representing multi-source data;

[0009] S4: Perform tensor completion on the singular vectors after singular value decomposition, and use evaluation indicators to evaluate the data after tensor completion;

[0010] S5: Extract data features based on the decomposed singular functions and the data after tensor completion, obtain the changing trends of pollutants at the detection point over time and space based on the data features, and obtain the monitoring results.

[0011] Preferably, the process of tensorizing the air quality data includes: the air quality tensor data is Where M and N represent the vertical and horizontal ranges of air quality data in geographical location, T represents the time range of air quality tensor data, and K represents the type of air quality data. The air quality data is converted into M×N×T×K Projected into the tensor space, the air quality tensor data is obtained; that is, when If there is observed data, the value at the corresponding position of Ω is 1, otherwise it is 0.

[0012] Preferably, the optimization problem of tensor completion is:

[0013]

[0014] Among them, λ l represents singular value, a l represents the singular vector, β l The vector representation of the spatial singular function representing the longitudinal range at the geographical location, γ l The vector representation of the spatial singular function representing the lateral extent at the geographic location, ξ l represents the vector representation of the time singular function, represents the air quality data tensor; Ω represents the index set, represents the linear projection operator under the sampling operation on Ω, represents the Tikhonov regularization term, represents the outer product of the vector, ∝ represents the weight of the regularization term, N represents the number of potential factor matrices, and U (n) represents the latent factor matrix, r represents the rank of the tensor, T represents the transpose, tr represents the trace of the matrix, represents the tensor component matrix, Similarity matrix representing the component matrices of a tensor.

[0015] Preferably, the process of performing singular value decomposition on tensor-represented data using a tensor singular value decomposition iterative update algorithm includes: converting a constrained objective function into a corresponding partially augmented Lagrangian objective function, and iteratively updating singular components using an alternating direction multiplier method; and using the updated singular components, completing and updating the tensor using a tensor completion formula.

[0016] Furthermore, the process of iteratively updating the singular components using the alternating direction multiplier method includes: obtaining a partial augmented Lagrangian function to represent the objective function; fixing other parameters in the objective function, and updating the parameter β according to the last iteration. l , γ l , l 、a l and l Solve the current parameter β in sequence l (t+1),γ l (t+1),ξ l (t+1), a l (t+1) and λ l (t+1); according to the updated parameters, the tensor is updated using the tensor completion formula; the updated tensor is tested to determine whether the current tensor meets the aggregation conditions. If the aggregation conditions are met, the tensor component (λ l ,β l ,λ l ,ξ l ,a l ) and the completed tensor If the aggregation condition is not met, continue to iterate and update the tensor component (λ l ,β l ,γ l ,ξ l ,a l ) and tensors Until the aggregation condition is met.

[0017] Furthermore, the tensor completion formula is:

[0018]

[0019] in, is the singular component (λ l ,βl ,γ l ,ξ l ,a l ) estimated air mass tensor; β l ∈R M is the vector representation of the spatial singular function of the longitudinal range at the geographical location, γ l ∈R N is the vector representation of the spatial singular function of the lateral range at the geographical location, ξ l ∈R T is the vector representation of the time singular function, a l ∈R M are singular vectors, is the outer product of the vectors, λ l is a singular value, and the tensor consists of r tensor components Each tensor component can be represented by its corresponding singular component.

[0020] Furthermore, the polymerization conditions are:

[0021]

[0022] Among them, Ω represents the index set, represents the linear projection operator under the sampling operation on Ω, Represents air quality tensor data, ∥.∥ F represents the F-norm, and ε represents the aggregation constant.

[0023] Preferably, using the evaluation index to evaluate the data after tensor completion includes using the square root error or the mean absolute error or the mean absolute percentage error to evaluate the data after tensor completion.

[0024] Preferably, the calculation formula of the root evaluation index includes:

[0025] The root mean square error formula is:

[0026]

[0027] The mean absolute error formula is:

[0028]

[0029] The mean absolute percentage error formula is:

[0030]

[0031] in, Represents the predicted value, y i Represents the true value, and m represents the total number of data after tensor completion.

[0032] A precise monitoring system for air pollutant concentration based on tensor representation, the system comprising: a data acquisition module, a data processing module, a tensor completion module and an output module;

[0033] The data acquisition module is used to acquire the air quality data of the point to be detected, and input the acquired air quality data into the data processing module;

[0034] The data processing module is used to represent the air quality number as tensor data and input the tensor data into the tensor completion module;

[0035] The tensor completion module uses a tensor completion formula to complete and update the tensor, and when the iteratively updated tensor components converge, the tensor data of the completed air quality is input;

[0036] The output module is used to obtain tensor data for completing air quality, and to visualize the obtained data to obtain output results.

[0037] Beneficial effects of the present invention:

[0038] 1. By fusing multi-source data and spatiotemporal air quality data, tensor completion is performed to obtain high-precision tensor data;

[0039] 2. Design a tensor singular function suitable for the time function pattern of the air quality data tensor representation, and extract the time-varying characteristics of the air quality data through tensor singular value decomposition;

[0040] 3. Design a tensor singular value function suitable for the spatial function pattern of the air quality data tensor representation, and include the spatially distributed air quality monitoring sites in the tensor representation range so that the tensor can represent spatial information;

[0041] 4. The tensor function pattern representation is divided into two parts: spatial function pattern and temporal function pattern, which correspond to the spatiotemporal characteristics of the tensor respectively, and the tensor spatiotemporal characteristics are characterized by integrating the tensor singular functions under these two function patterns;

[0042] 5. Representing air quality data as a tensor can describe the data more concisely. According to the properties of the tensor, the changes in the air quality of the target city can be observed and analyzed from multiple angles. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 The flowchart of the method for accurately monitoring the concentration of air pollutants based on tensor representation of the present invention is as follows;

[0044] Figure 2 A visualization structure diagram of a sub-tensor of the air quality tensor of the present invention;

[0045] Figure 3 The air quality data distribution diagram after tensor completion of the present invention;

[0046] Figure 4 A schematic diagram of a system for visualizing air quality tensor data of the present invention;

[0047] Figure 5 A schematic diagram of a system for cross-sectional visualization of air quality tensor data of the present invention;

[0048] Figure 6 A schematic diagram of a system for singular component analysis of air quality tensor data according to the present invention. DETAILED DESCRIPTION

[0049] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0050] The present invention aims to use tensor-represented air pollutant concentrations and tensor singular value decomposition algorithms to achieve spatiotemporal gridded monitoring of urban air pollutants. The present invention uses tensor representation to describe spatially sparse air quality data, and uses tensors with functional patterns in time and space to decompose air quality tensor data to obtain tensor singular vectors and singular functions. Tensor components can not only be used to reconstruct air quality tensors to obtain air quality data with high spatial resolution, but the singular functions obtained by tensor decomposition are also an important indicator for analyzing tensors.

[0051] A method for accurately monitoring air pollutant concentration based on tensor representation, the method comprising: acquiring air quality data of a point to be detected in real time, inputting the acquired air quality data into a monitoring model to obtain monitoring results; and marking the monitoring point according to the monitoring results.

[0052] like Figure 1 As shown in Figure 1, the process of using the monitoring model to process air quality data includes:

[0053] S1: The air quality data is represented by tensor; the tensor representation of the monitoring point includes the spatial distribution of the station, the time series of pollutant concentration, and the pollutant concentration;

[0054] S2: Construct a tensor completion optimization problem based on the tensor representation of air quality;

[0055] S3: According to the tensor completion optimization problem, the tensor singular value decomposition iterative update algorithm is used to perform singular value decomposition on the tensor representation data to obtain spatial singular functions, temporal singular functions and singular vectors representing multi-source data;

[0056] S4: Use the singular vectors and singular functions after singular value decomposition to complete the tensor, and use evaluation indicators to evaluate the data after tensor completion;

[0057] S5: Extract data features based on the decomposed singular functions and the data after tensor completion, obtain the changing trends of pollutants at the detection point over time and space based on the data features, and obtain the monitoring results.

[0058] The process of tensorizing air quality data includes: the air quality tensor data is Where M and N represent the vertical and horizontal ranges of air quality data in geographical location, T represents the time range of air quality tensor data, and K represents the type of air quality data. The air quality data is converted into M×N×T×K Projected into the tensor space, the air quality tensor data is obtained; that is, when If there is observed data, the value at the corresponding position of Ω is 1, otherwise it is 0.

[0059] The optimization problem of tensor completion is:

[0060]

[0061] Where W represents the weight matrix, represents the air quality tensor data, λ l represents singular value, a l represents the singular vector, β l The vector representation of the spatial singular function representing the longitudinal range at the geographical location, γ l The vector representation of the spatial singular function representing the lateral extent at the geographic location, ξ l represents the vector representation of the time singular function, Represents the air quality data tensor; represents the Tikhonov regularization term, which is used to constrain the latent factor matrix U (n) , to prevent overfitting and ensure the uniqueness of the solution ∝ is the weight of the regularization term, and N is the number of potential factor matrices; is the tensor component matrix, which is the matrix obtained by the outer product of the tensor singular vectors. Assume that the relationship between the potential factor matrix and the tensor component matrix is That is, the potential factor of the space is characterized by the spatial tensor function, and α is used to control The weight of It is a similarity matrix that describes the tensor component matrix. For example, the similarity matrix of the spatial tensor component matrix represents the spatial variation constraints of air quality data.

[0062] The process of performing singular value decomposition on tensor-represented data using a tensor singular value decomposition iterative update algorithm includes: converting a constrained objective function into a corresponding partially augmented Lagrangian objective function, iteratively updating singular components using an alternating direction multiplier method; and completing and updating the tensor using a tensor completion formula based on the updated singular components.

[0063] The process of iteratively updating the singular components using the alternating direction multiplier method includes: obtaining a partial augmented Lagrangian function to represent the objective function; fixing other parameters in the objective function, and updating the parameters β according to the last iteration. l , γ l , l 、a l and l Solve the current parameter β in sequence l (t+1),γ l (t+1),ξ l (t+1), a l (t+1) and λ l (t+1); according to the updated parameters, the tensor is updated using the tensor completion formula; the updated tensor is tested to determine whether the current tensor meets the aggregation conditions. If the aggregation conditions are met, the tensor component (λ l ,β l ,γ l ,ξ l ,a l ) and the completed tensor If the aggregation condition is not met, continue to iterate and update the tensor component (λ l ,β l ,γ l ,ξ l ,a l ) and tensors Until the aggregation condition is met.

[0064] The polymerization conditions are:

[0065]

[0066] Among them, Ω represents the index set, represents the linear projection operator under the sampling operation on Ω, Represents air quality tensor data, i.e., an observation tensor with missing values; ∥.∥ F represents the F-norm, and ε represents the aggregation constant.

[0067] The process of tensor completion of the singular vector after singular value decomposition includes: obtaining the updated singular components; completing the tensor data using the tensor completion formula according to the updated singular components to obtain the completed tensor data. The tensor completion formula is:

[0068]

[0069] in, is the singular component (λ l ,β l ,γ l ,ξ l ,a l ) estimated air mass tensor; β l ∈R M is the vector representation of the spatial singular function of the longitudinal range at the geographical location, γ l ∈R N is the vector representation of the spatial singular function of the lateral range at the geographical location, ξ l ∈R T is the vector representation of the time singular function, a l ∈R M are singular vectors, is the outer product of the vectors, λ l is a singular value, and the tensor consists of r tensor components Each tensor component can be represented by its corresponding singular component.

[0070] A specific implementation of a method for accurately monitoring air pollutant concentration based on tensor representation, the method comprising:

[0071] Step 1: Tensor representation of air quality data. Air quality monitoring stations are sparsely distributed in the city's geographic space. Air quality monitoring stations will obtain real-time concentrations of various pollutants in the air, including PM2.5, PM10, ozone and other data. The spatial distribution of air quality monitoring stations, the time series of pollutant concentrations and the concentrations of various air pollutants in the air are characterized by a tensor to obtain a fourth-order air quality tensor.

[0072] Step 2, construct the tensor singular function of the air quality tensor. Since the pollutant concentration in the air quality is stored in a table mode, the representation of the air quality monitoring site in the geographic space needs to be described by a function mode, which is called a spatial singular function. The spatial singular function is a function that describes the relationship between the longitude and latitude of the site and the air quality. The time function mode is a time singular function used to describe the time characteristics of air quality data. Since the spatial singular function and the time singular function are continuous, and the collected observations are discrete, it is necessary to describe the singular function by a tensor composed of discrete observation data.

[0073] Step 3, design an iterative update algorithm for tensor singular value decomposition; the designed update algorithm is to select the tensor decomposition algorithm as CP decomposition, and define the result of CP decomposition, including a time function component, two space function components and a singular vector. Since tensor decomposition contains tensor products and multilinear structures, this will make the model problem highly non-convex or even NP-difficult, and solving the tensor singular function is an infinite-dimensional optimization problem, so it is necessary to introduce solution constraints and design iterative algorithms to ensure the smooth progress of iterative solutions. The constraints are constraints on solving time functions and constraints on introducing prior knowledge of changes in air pollutant concentrations in geographical locations. The principles of iterative update algorithms for time functions and space functions are different. Among them, iterative update solutions to time functions are essentially to solve the minimizer of the regularized empirical risk function defined on the reproducing kernel Hilbert space, while the solutions to the two space functions are to solve the continuous functions estimated by discrete data that change the air quality data with longitude and latitude.

[0074] Step 4: Decompose the singular values ​​of the tensor using the tensor singular value decomposition iterative update algorithm. The tensor singular value decomposition result includes two spatial singular functions, corresponding to the longitude and latitude of the monitoring site, a temporal singular function representing the temporal characteristics of air quality, and a singular vector representing multi-source data.

[0075] Step 5, perform tensor completion on the singular vectors after singular value decomposition. The tensor before decomposition is sparse, and the tensor components after reconstructing the decomposition can be completed. The completed tensor is dense. Before tensor decomposition, some data can be "erased" and their positions in the tensor are recorded; the "erased" data is randomly selected from the tensor data, accounting for 10% or 30% of all data. After tensor completion, the completed tensor is compared with the test data, and the effect of tensor completion can be tested by evaluation indicators such as root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE).

[0076] The root mean square error formula is:

[0077]

[0078] The mean absolute error formula is:

[0079]

[0080] The mean absolute percentage error formula is:

[0081]

[0082] in, Represents the predicted value, y i Represents the true value, and m represents the total number of data after tensor completion.

[0083] Step S6, visualization of the completed tensor and singular functions and singular vectors. The singular functions and singular vectors obtained by tensor decomposition imply the potential data organization characteristics inside the tensor. For example: analyzing the relationship between temporal singular functions and singular vectors to obtain the characteristics and trends of air pollutant concentrations changing over time; analyzing the relationship between spatial singular functions and singular vectors to obtain the geographical distribution of air pollutant concentrations and predict the trend of air pollutant concentrations. A precise air pollutant concentration monitoring system based on tensor representation is used to visualize air quality tensor data, output a tensor representation of air quality data, and arbitrarily select a tensor section for analysis. Singular functions and singular vectors can intuitively show researchers the changing patterns of air quality data by drawing curve graphs.

[0084] like Figure 2 The sub-tensor visualization of the air mass tensor is shown. Because the air mass tensor is a fourth-order tensor and the fourth-order tensor cannot be represented by a three-dimensional graph, only the third-order sub-tensor of the air mass tensor is shown here.

[0085] Figure 3 It is the air quality data after tensor completion. The observation data is often sparsely distributed in the geographical space of the monitored city, but after tensor completion, the distribution of air quality data at each moment in the geographical space becomes dense.

[0086] Figure 4 This is a system diagram for visualizing air quality tensor data. Air quality tensor data is intuitively displayed in front of the operator. In this interface, you can choose the tensor display method, including the selection of slices, the selection of tensor data types, etc.

[0087] Figure 5 It is a system diagram for cross-sectional visualization of air quality tensor data. By expanding the air quality tensor data in a time mode, the distribution of air quality data in geographic space at each moment can be observed, which will be beneficial for the air quality center to release the city's air quality status.

[0088] Figure 6 This is a system diagram of singular component analysis of air quality tensor data. By visualizing singular components and plotting longitude singular functions and latitude singular functions on a graph, we can see the changing trends of these two singular components over time. Visual analysis of singular components is helpful to reveal the potential laws of urban air quality changes.

[0089] A precise monitoring system for air pollutant concentration based on tensor representation, the system comprising: a data acquisition module, a data processing module, a tensor completion module and an output module;

[0090] The data acquisition module is used to acquire the air quality data of the point to be detected, and input the acquired air quality data into the data processing module;

[0091] The data processing module is used to represent the air quality number as tensor data and input the tensor data into the tensor completion module;

[0092] The tensor completion module uses a tensor completion formula to complete and update the tensor, and when the iteratively updated tensor components converge, the tensor data of the completed air quality is input;

[0093] The output module is used to obtain tensor data for completing air quality, and to visualize the obtained data to obtain output results.

[0094] The specific implementation of the system of the present invention is the same as the specific implementation of the method of the present invention.

[0095] The above embodiments further illustrate the purpose, technical solutions and advantages of the present invention in detail. It should be understood that the above embodiments are only preferred implementation modes of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made to the present invention within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for accurately monitoring air pollutant concentration based on tensor representation. It is characterized in that include: Acquire the air quality data of the monitoring point in real time, input the acquired air quality data into the monitoring model, and obtain the monitoring results; Mark the monitoring point according to the monitoring results; The process of processing air quality data using the monitoring model includes: S1: The air quality data is represented by tensor; the tensor representation of the monitoring point includes the spatial distribution of the station, the time series of pollutant concentration, and the pollutant concentration; S2: Based on the tensor representation of air quality, a tensor completion optimization problem is constructed; the tensor completion optimization problem is: Among them, λ l represents singular value, a l represents the singular vector, β l The vector representation of the spatial singular function representing the longitudinal range at the geographical location, γ l The vector representation of the spatial singular function representing the lateral extent at the geographic location, ξ l represents the vector representation of the time singular function, x represents the air quality data tensor; Ω represents the index set, represents the linear projection operator under the sampling operation on Ω, represents the Tikhonov regularization term, represents the outer product of the vector, ∝ represents the weight of the regularization term, N represents the number of potential factor matrices, and U (n) represents the latent factor matrix, r represents the rank of the tensor, T represents the transpose, tr represents the trace of the matrix, represents the tensor component matrix, Similarity matrix representing the tensor component matrix; S3: According to the tensor completion optimization problem, the tensor singular value decomposition iterative update algorithm is used to perform singular value decomposition on the tensor representation data to obtain spatial singular functions, temporal singular functions and singular vectors representing multi-source data; S4: Use the singular vectors and singular functions after singular value decomposition to complete the tensor, and use evaluation indicators to evaluate the data after tensor completion; S5: Extract data features based on the decomposed singular functions and the data after tensor completion, obtain the changing trends of pollutants at the detection point over time and space based on the data features, and obtain the monitoring results.

2. According to claim 1, a method for accurately monitoring air pollutant concentration based on tensor representation, It is characterized in that The process of tensorizing air quality data includes: the air quality tensor data is Where M and N represent the vertical and horizontal ranges of the air quality data in terms of geographical location, T represents the time range of the air quality tensor data, and K represents the type of air quality data. The air quality data is indexed by the index set Ω∈R M×N×T×K Projected into the tensor space, the air quality tensor data is obtained; that is, when If there is observed data, the value at the corresponding position of Ω is 1, otherwise it is 0.

3. According to claim 1, a method for accurately monitoring air pollutant concentration based on tensor representation, It is characterized in that The process of performing singular value decomposition on tensor-represented data using a tensor singular value decomposition iterative update algorithm includes: converting a constrained objective function into a corresponding partially augmented Lagrangian objective function, iteratively updating singular components using an alternating direction multiplier method; and completing and updating the tensor using a tensor completion formula based on the updated singular components.

4. According to claim 3, a method for accurately monitoring air pollutant concentration based on tensor representation, It is characterized in that The process of iteratively updating the singular components using the alternating direction multiplier method includes: obtaining a partial augmented Lagrangian function to represent the objective function; fixing other parameters in the objective function, and updating the parameters β according to the last iteration. l , γ l , l 、a l and l Solve the current parameter β in sequence l (t+1),γ l (t+1),ξ l (t+1), a l (t+1) and λ l (t+1); according to the updated parameters, the tensor is updated using the tensor completion formula; the updated tensor is tested to determine whether the current tensor meets the aggregation conditions. If the aggregation conditions are met, the tensor component (λ l , β l , γ l ,ξ l , a l ) and the completed tensor χ; if the aggregation condition is not met, continue to iterate and update the tensor component (λ l , β l , γ l ,ξ l , a l ) and the tensor χ until the aggregation condition is met.

5. According to claim 4, a method for accurately monitoring air pollutant concentration based on tensor representation, It is characterized in that The tensor completion formula is: in, is the singular component (λ l , β l , γ l ,ξ l , a l ) estimated air mass tensor; β l ∈R M is the vector representation of the spatial singular function of the longitudinal range at the geographical location, γ l ∈R N is the vector representation of the spatial singular function of the lateral range at the geographical location, ξ l ∈R T is the vector representation of the time singular function, a l ∈R M are singular vectors, is the outer product of the vectors, λ l is a singular value; M and N represent the vertical and horizontal ranges of the air quality data in terms of geographical location, respectively; T represents the time range of the air quality tensor data; and K represents the type of air quality data.

6. According to claim 4, a method for accurately monitoring air pollutant concentration based on tensor representation, It is characterized in that The polymerization conditions are: Among them, Ω represents the index set, represents the linear projection operator under the sampling operation on Ω, Represents the air quality tensor data, ||.|| F represents the F-norm, and ε represents the aggregation constant.

7. According to claim 1, a method for accurately monitoring air pollutant concentration based on tensor representation, It is characterized in that The evaluation indicators used to evaluate the data after tensor completion include using the square root error or the mean absolute error or the mean absolute percentage error to evaluate the data after tensor completion.

8. According to claim 7, a method for accurately monitoring air pollutant concentration based on tensor representation, It is characterized in that The calculation formulas for the evaluation indicators include: The root mean square error formula is: The mean absolute error formula is: The mean absolute percentage error formula is: in, It represents the predicted value of the data after tensor completion, yi represents the true value of the data, and m represents the total number of data after tensor completion.

9. A system for accurately monitoring air pollutant concentration based on tensor representation, the system being used to execute the method for accurately monitoring air pollutant concentration based on tensor representation according to claim 1, It is characterized in that The system comprises: a data acquisition module, a data processing module, a tensor completion module and an output module; The data acquisition module is used to acquire the air quality data of the point to be detected, and input the acquired air quality data into the data processing module; The data processing module is used to represent the air quality number as tensor data and input the tensor data into the tensor completion module; The tensor completion module uses a tensor completion formula to complete and update the tensor, and when the iteratively updated tensor components converge, the tensor data of the completed air quality is input; The output module is used to obtain tensor data for completing air quality, and to visualize the obtained data to obtain output results.

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