A spatiotemporal data completion method based on composite non-convex function and structural regression

Abstract

The present invention belongs to the field of spatiotemporal data completion technology, and specifically relates to a spatiotemporal data completion method based on composite non-convex functions and structural regression, the steps of which include storing the spatiotemporal data to be completed as a target tensor in the form of a three-dimensional tensor; constructing the WCT norm of the target tensor to capture the global multidimensional low-rank characteristics of the spatiotemporal data in the transform domain; constructing a structural regression model of the target tensor to reveal complex spatiotemporal data related patterns; combining the WCT norm and the structural regression model to construct a spatiotemporal data completion model, and introducing auxiliary variables and additional constraints to transform the objective function of the spatiotemporal data completion model; using the alternating direction multiplier method to solve the transformed objective function, obtaining the optimal solution of the target tensor, and completing the completion of the spatiotemporal data. The present invention effectively captures the global multidimensional low-rank properties of tensors in the transform domain, and improves the effect of tensor recovery.

Description

Technical Field

[0001] The present invention belongs to the technical field of spatiotemporal data completion, and in particular relates to a spatiotemporal data completion method based on composite non-convex functions and structural regression. Background Art

[0002] With the rapid development of the Internet of Things, big data, and artificial intelligence technologies, spatiotemporal data has been widely used in urban planning, environmental monitoring, traffic management, weather forecasting, and other fields. These data usually have temporal and spatial correlations, but due to sensor failures, network interruptions, data collection conditions, and other reasons, there are often missing values ​​in spatiotemporal data. These missing values ​​not only affect the integrity of the data, but may also lead to a decline in the performance of the analysis model and even lead to wrong decisions.

[0003] At present, the research on missing value completion of spatiotemporal data mainly focuses on the following methods: interpolation methods based on statistical models (such as linear interpolation and Kriging interpolation), regression methods based on machine learning (such as support vector machines and decision trees), and models based on deep learning (such as recurrent neural networks and graph neural networks). However, these methods still have certain limitations when dealing with large-scale, high-dimensional, and complex-correlated spatiotemporal data.

[0004] In recent years, matrix and tensor completion methods have been successfully applied to spatiotemporal data recovery. These methods treat spatiotemporal data as two-dimensional matrices or high-order tensors and use the inherent low-rank structure of the data to estimate missing values. Compared with low-rank matrix completion (LRMC), low-rank tensor completion (LRTC) can better preserve the multidimensional structure of spatiotemporal data, thereby achieving better recovery performance. However, the rank minimization problem in LRMC and LRTC is NP-hard (NP refers to a class of problems whose solutions can be verified in polynomial time, but it is not known whether there is a polynomial time algorithm to solve these problems) and is difficult to solve. Summary of the invention

[0005] In view of the above deficiencies in the prior art, the purpose of the present invention is to provide a spatiotemporal data completion method based on composite non-convex functions and structural regression, which effectively captures the global multidimensional low-rank properties of tensors in the transform domain and improves the effect of tensor recovery.

[0006] To achieve the above objectives, the present invention provides a spatiotemporal data completion method based on composite non-convex function and structural regression, comprising the following steps:

[0007] S1. Store the spatiotemporal data to be completed as a target tensor in the form of a three-dimensional tensor;

[0008] S2, construct the WCT norm of the target tensor to capture the global multi-dimensional low-rank characteristics of spatiotemporal data in the transform domain;

[0009] S3, constructing a structural regression model of the target tensor to reveal complex spatiotemporal data correlation patterns;

[0010] S4. Combining the WCT norm and the structural regression model, a spatiotemporal data completion model is constructed, and auxiliary variables and additional constraints are introduced to transform the objective function of the spatiotemporal data completion model;

[0011] S5. Use the alternating direction multiplier method to solve the transformed objective function, obtain the optimal solution of the target tensor, and complete the completion of the spatiotemporal data.

[0012] As a preferred solution of the present invention, in S1, the process of storing the spatiotemporal data to be completed as a target tensor in the form of a three-dimensional tensor is:

[0013] S1.1. Represent spatiotemporal data with missing data as a three-dimensional tensor , is the real data tensor, where express dimensional real vector space, n1 represents the number of devices used to collect data (such as detector devices), n2 represents the number of time periods for collecting data on each natural day, and n3 represents the number of days for collecting data;

[0014] S1.2, use express The index of the observation in , correspondingly, is express The index of missing data in;

[0015] S1.3. Constructing a three-dimensional tensor of spatiotemporal data to be repaired , X is the target tensor, The observations in are transferred to the corresponding positions in X.

[0016] As a preferred solution of the present invention, in S2, the WCT norm of the target tensor It is expressed as:

[0017] (1);

[0018] In the formula, w u is a non-negative weight coefficient, u represents the index of the mode of the tensor, u=1, 2, 3, and w1+w2+w3=1; represents the tensor transformation of X on the u-th mode; for The lth slice matrix, the rest of the superscript Similarly ( etc.), n u is the number of slice matrices; express The i-th singular value of , m is the number of singular values; is a composite non-convex function.

[0019] As a preferred embodiment of the present invention, in said S2, Specifically expressed as:

[0020] (2);

[0021] Where x represents the input of the composite non-convex function; p is a non-negative parameter, 0<p≤1.

[0022] As a preferred embodiment of the present invention, in S3, the structural regression model is expressed as:

[0023] (3);

[0024] In the formula, , , , They are X, P, , The expansion matrix on the second mode, the remaining subscripts Similarly ( etc.); H is the autoregressive coefficient matrix; represents the Hadamard product; Z represents the sample coefficient matrix; represents the random error matrix.

[0025] As a preferred embodiment of the present invention, H is represented by:

[0026] (4).

[0027] As a preferred embodiment of the present invention, in S4, the method for constructing the spatiotemporal data completion model and transforming the objective function of the spatiotemporal data completion model is:

[0028] S4.1. The constructed spatiotemporal data completion model is expressed as:

[0029] (5);

[0030] In the formula, , , is a non-negative parameter; the subscript F represents the Frobenius norm; the subscript 1 represents the L1 norm;

[0031] The constraints are:

[0032] (6);

[0033] In the formula, represents the diagonal matrix of Z, and the rest Similarly;

[0034] S4.2. Introducing auxiliary variables M1, M2, M3, D, G, U and V, formula (5) is transformed into the following objective function:

[0035] (7);

[0036] In the formula, Indicates M u The tensor transformation on the u-th mode, M u That is, M1, M2, M3; the subscript WCT represents the WCT norm; the constraints of each auxiliary variable are: M u =X,D=X, , , ;

[0037] The constraints for the target tensor X are:

[0038] (8);

[0039] S4.3, the augmented Lagrangian function corresponding to equation (7) is expressed as:

[0040] (9);

[0041] In the formula, y u , y4, Y5, Y6, and Y7 are all Lagrange multipliers; is the penalty parameter.

[0042] As a preferred embodiment of the present invention, in S5, the solution process is:

[0043] S5.1. Input non-negative weight coefficient w u , non-negative parameter , , , iteration stop parameter , and the maximum number of iterations ;

[0044] S5.2. Initialize parameters to get the initial value X of the target tensor 0 , the initial value of the sample coefficient matrix Z 0 , initial value of auxiliary variable G 0 、V 0 , U 0 , the initial value of the Lagrange multiplier , ,as well as , , ;

[0045] S5.3, iteratively update each variable in S5.2 using the alternating direction multiplication method;

[0046] S5.4, determine whether the iteration stop condition is met, if the iteration stop condition is met, stop the iteration, otherwise update k=k+1, k represents the number of iterations, and return to S5.3 to continue a new round of iteration;

[0047] S5.5. After the iteration stops, the final completed tensor is output, which is the target tensor X, to obtain the final completed spatiotemporal data.

[0048] As a preferred solution of the present invention, in S5.3, the process of iteratively updating the variables using the alternating direction multiplier method is:

[0049] S5.3.1. Update auxiliary variable M u , the process is as follows:

[0050] S5.3.1.1, respectively and Do tensor singular value decomposition t-SVD, , , are respectively , X, y u , the superscript k indicates the kth iteration, and the same is true for the other superscript k (including V K , U K wait); Indicated in Tensor transformation on the u-th mode;

[0051] After decomposition, we get the corresponding and , where Q1, Q2, R1, R2 are orthogonal tensors; and are the conjugate transposes of R1 and R2 respectively; S1 and S2 are f-diagonal tensors;

[0052] S5.3.1.2. Calculation of the k+1th iteration , that is , the superscript k+1 indicates the k+1th iteration, and the same applies to the other superscript k+1s (including , etc.), calculate The formula is as follows:

[0053] (10);

[0054] In the formula, is an f-diagonal tensor satisfying:

[0055] (11);

[0056] In the formula, Represents the element in S1 with index (a, b, j), 1≤a≤ , 1≤b≤ , 1≤j≤ , the rest is similar ( wait); is the threshold value, ; yes The derivative of is expressed as:

[0057] (12);

[0058] Where x represents the input of the composite non-convex function; p is a non-negative parameter, 0<p≤1;

[0059] S5.3.1.3, Yes Perform the inverse tensor transformation on the u-th mode to obtain M at the k+1th iteration u , that is ;

[0060] S5.3.2. Update the auxiliary variable D. The formula is as follows:

[0061] (13);

[0062] In the formula, represents the inverse operation of the second mode expansion;

[0063] S5.3.3. Update the auxiliary variable G. The formula is as follows:

[0064] (14);

[0065] In the formula, Represents element-wise division of two matrices; is a matrix whose elements are all 1;

[0066] S5.3.4. Update the auxiliary variable V. The formula is as follows:

[0067] (15);

[0068] Where I is the identity matrix; the superscript T indicates transpose;

[0069] S5.3.5. Update the auxiliary variable U. The formula is as follows:

[0070] (16);

[0071] Where A is an intermediate matrix, ;

[0072] S5.3.6. Update the auxiliary variable Z. The formula is as follows:

[0073] (17);

[0074] In the formula, J is the intermediate variable, , is the intermediate variable, ;

[0075] S5.3.7. Update the target tensor X. The formula is as follows:

[0076] (18);

[0077] In the formula, B is an intermediate variable, expressed as:

[0078] (19);

[0079] S5.3.8. Update the Lagrange multiplier as follows:

[0080] (20);

[0081] In the formula, Indicates update.

[0082] As a preferred solution of the present invention, in S5.4, the iteration stop condition is: or When , stop the iteration.

[0083] The algorithm involved in the present invention can be executed by an electronic device, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor, and the above algorithm calculation is implemented by executing the software by the processor.

[0084] The beneficial effects of the present invention are:

[0085] The present invention effectively captures the global multidimensional low-rank properties of tensors in the transform domain by constructing a new non-convex proxy rank function and weighted composite tensor (WCT) norm, which can estimate the tensor rank more accurately than the traditional tensor kernel norm. Secondly, the designed structural regression model can simultaneously express the local temporal smoothness and sample similarity of spatiotemporal data, so that the recovered tensor can retain the local smoothness of the tensor along the time dimension and the similarity of the spatial dimension while maintaining the global low rank.

[0086] The present invention uses the alternating direction method of multipliers (ADMM) to optimize and solve the proposed low-rank tensor completion model. Each subproblem can be solved by a closed-form equation, which further improves the effect of tensor recovery compared with existing methods. This method shows high recovery performance when processing large-scale, high-dimensional, and complex-correlated spatiotemporal data. In particular, the experimental results on the PeMS dataset show that the present invention can achieve a low root mean square error (RMSE) under different missing modes, proving its effectiveness and superiority in practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] Figure 1 It is a schematic diagram of the process of the present invention;

[0088] Figure 2 It is a schematic diagram of the convergence curve in the absence of MCAR in the verification process of the present invention. DETAILED DESCRIPTION

[0089] The embodiments of the present invention are further described below in conjunction with the accompanying drawings:

[0090] like Figure 1 As shown, a spatiotemporal data completion method based on composite non-convex function and structural regression includes the following steps:

[0091] S1. Store the spatiotemporal data to be completed as a target tensor in the form of a three-dimensional tensor;

[0092] S2, construct the WCT norm of the target tensor to capture the global multi-dimensional low-rank characteristics of spatiotemporal data in the transform domain;

[0093] S3, constructing a structural regression model of the target tensor to reveal complex spatiotemporal data correlation patterns;

[0094] S4. Combining the WCT norm and the structural regression model, a spatiotemporal data completion model is constructed, and auxiliary variables and additional constraints are introduced to transform the objective function of the spatiotemporal data completion model;

[0095] S5. Use the alternating direction multiplier method to solve the transformed objective function, obtain the optimal solution of the target tensor, and complete the completion of the spatiotemporal data.

[0096] A tensor is a multidimensional array that can have any number of dimensions.

[0097] In S1, the process of storing the spatiotemporal data to be completed as a target tensor in the form of a three-dimensional tensor is:

[0098] S1.1. Represent spatiotemporal data with missing data as a three-dimensional tensor , is the real data tensor, where express dimensional real vector space, n1 represents the number of devices used to collect data, n2 represents the number of time periods for collecting data on each natural day, and n3 represents the number of days for collecting data;

[0099] S1.2, use express The index of the observation in , correspondingly, is express The index of missing data in;

[0100] S1.3. Constructing a three-dimensional tensor of spatiotemporal data to be repaired , X is the target tensor, The observations in are transferred to the corresponding positions in X.

[0101] In S2, the WCT norm of the target tensor It is expressed as:

[0102] (1);

[0103] In the formula, w u is a non-negative weight coefficient, u represents the index of the mode of the tensor, u=1, 2, 3, and w1+w2+w3=1; represents the tensor transformation of X on the u-th mode; for The lth slice matrix, the rest of the superscript Similarly, n u is the number of slice matrices; express The i-th singular value of , m is the number of singular values; is a composite non-convex function.

[0104] Singular value is a concept in linear algebra and is part of the singular value decomposition (SVD) of a matrix or tensor.

[0105] In S2, Specifically expressed as:

[0106] (2);

[0107] Where x represents the input of the composite non-convex function; p is a non-negative parameter, 0<p≤1.

[0108] In S3, the structural regression model is expressed as:

[0109] (3);

[0110] In the formula, , , , They are X, P, , The expansion matrix on the second mode, the remaining subscripts Similarly; H is the autoregressive coefficient matrix; represents the Hadamard product; Z represents the sample coefficient matrix; represents the random error matrix.

[0111] H is represented by:

[0112] (4).

[0113] In S4, the method for constructing the spatiotemporal data completion model and transforming the objective function of the spatiotemporal data completion model is:

[0114] S4.1. The constructed spatiotemporal data completion model is expressed as:

[0115] (5);

[0116] In the formula, , , is a non-negative parameter; the subscript F represents the Frobenius norm (a type of matrix norm); the subscript 1 represents the L1 norm (also called the column sum norm or Manhattan norm);

[0117] The constraints are:

[0118] (6);

[0119] In the formula, represents the diagonal matrix of Z, and the rest Similarly;

[0120] S4.2. Introducing auxiliary variables M1, M2, M3, D, G, U and V, formula (5) is transformed into the following objective function:

[0121] (7);

[0122] In the formula, Indicates M u The tensor transformation on the u-th mode, M u That is, M1, M2, M3; the subscript WCT represents the WCT norm; the constraints of each auxiliary variable are: M u =X,D=X, , , ;

[0123] The constraints for the target tensor X are:

[0124] (8);

[0125] S4.3, the augmented Lagrangian function corresponding to equation (7) is expressed as:

[0126] (9);

[0127] In the formula, y u , y4, Y5, Y6, and Y7 are all Lagrange multipliers; is the penalty parameter.

[0128] Penalty parameter is an important concept in mathematical optimization and computational methods, especially in constrained optimization problems. Penalty parameter is used to incorporate constraints into the objective function, thus transforming the constrained optimization problem into an unconstrained optimization problem.

[0129] In S5, the solution process is:

[0130] S5.1. Input non-negative weight coefficient w u , non-negative parameter , , , iteration stop parameter , and the maximum number of iterations ;

[0131] S5.2. Initialize parameters to get the initial value X of the target tensor 0 , the initial value of the sample coefficient matrix Z 0 , initial value of auxiliary variable G 0 、V 0 , U 0 , the initial value of the Lagrange multiplier , ,as well as , , ;

[0132] S5.3, iteratively update each variable in S5.2 using the alternating direction multiplication method;

[0133] S5.4, determine whether the iteration stop condition is met, if the iteration stop condition is met, stop the iteration, otherwise update k=k+1, k represents the number of iterations, and return to S5.3 to continue a new round of iteration; the iteration stop condition is: when or When , stop the iteration;

[0134] S5.5. After the iteration stops, the final completed tensor is output, which is the target tensor X, to obtain the final completed spatiotemporal data.

[0135] In S5.3, the process of iteratively updating variables using the alternating direction multiplier method is:

[0136] S5.3.1. Update auxiliary variable M u , the process is as follows:

[0137] S5.3.1.1, respectively and Do tensor singular value decomposition t-SVD, , , are respectively , X, y u , the superscript k indicates the kth iteration, and the other superscript k are similar; Indicated in Tensor transformation on the u-th mode;

[0138] Tensor Singular Value Decomposition (t-SVD) decomposes a high-order tensor into a series of matrices and vector products, similar to the singular value decomposition (SVD) of a matrix, and is represented by tSVD.

[0139] After decomposition, we get the corresponding and , where Q1, Q2, R1, R2 are orthogonal tensors; and are the conjugate transposes of R1 and R2 respectively; S1 and S2 are f-diagonal tensors;

[0140] An f-diagonal tensor is a special type of tensor where nonzero elements appear only when all modes (or dimensions) have the same index;

[0141] S5.3.1.2. Calculation of the k+1th iteration , that is , the superscript k+1 indicates the k+1th iteration, and the other superscript k+1 is the same, calculation The formula is as follows:

[0142] (10);

[0143] In the formula, is an f-diagonal tensor satisfying:

[0144] (11);

[0145] In the formula, Represents the element in S1 with index (a, b, j), 1≤a≤ , 1≤b≤ , 1≤j≤ , the rest are similar; is the threshold value, ; yes The derivative of is expressed as:

[0146] (12);

[0147] S5.3.1.3, Yes Perform the inverse tensor transformation on the u-th mode to obtain M at the k+1th iteration u , that is ;

[0148] S5.3.2. Update the auxiliary variable D. The formula is as follows:

[0149] (13);

[0150] In the formula, represents the inverse operation of the second mode expansion;

[0151] S5.3.3. Update the auxiliary variable G. The formula is as follows:

[0152] (14);

[0153] In the formula, Represents element-wise division of two matrices; is a matrix whose elements are all 1;

[0154] S5.3.4. Update the auxiliary variable V. The formula is as follows:

[0155] (15);

[0156] Where I is the identity matrix; the superscript T indicates transpose;

[0157] S5.3.5. Update the auxiliary variable U. The formula is as follows:

[0158] (16);

[0159] Where A is an intermediate matrix, ;

[0160] S5.3.6. Update the auxiliary variable Z. The formula is as follows:

[0161] (17);

[0162] In the formula, J is the intermediate variable, , is the intermediate variable, ;

[0163] S5.3.7. Update the target tensor X. The formula is as follows:

[0164] (18);

[0165] In the formula, B is an intermediate variable, expressed as:

[0166] (19);

[0167] S5.3.8. Update the Lagrange multiplier as follows:

[0168] (20);

[0169] In the formula, Indicates update.

[0170] The validation process is to select spatiotemporal data containing missing data. The dataset used is PeMS, which contains 40 days of highway traffic speed information collected by 30 traffic sensors in a certain area. The collection interval is 5 minutes, and 288 data are generated every day. 50% of the observations are masked as missing values, and the simulated incomplete spatiotemporal data are used as input. Three common missing modes are simulated, including completely missing at random (MCAR), missing at random (MAR), and mixed missing (MIX).

[0171] Based on the method of this embodiment, data completion is performed. In the case of MCAR, for w u , w1=0.73705, w2=0.14767, non-negative parameter p=0.95118. In the MIX case, w1=0.68665, w2=0.09770, p=0.96280. In the MAR case, w1=0.78326, w2=0.00588, p=0.98997; w3=1-w1-w2.

[0172] In the case of MCAR, =1.98901, =15.75073, =0.20681. In the case of MIX, =4.49072, =23.07618, =0.23979. In the case of MAR, =8.88354, =13.96606, =0.38013.

[0173] Calculate the initial value of the tensor to be completed by initializing the value of the missing position to the average value of the observed value , , , , , the initial value of the Lagrange multiplier , , , .

[0174] In order to measure the completion capability of the present invention, the root mean square error RMSE is defined as:

[0175] (twenty one);

[0176] Where, the number of missing values ​​is C; the set of indices of all missing values ​​is ; , Represent the true value and the predicted value respectively. Based on the method of this embodiment, the RMSE under the three missing modes MCAR, MIX and MAR are 1.960, 2.507 and 2.898 respectively, which has a high recovery performance. Among them, in the case of MCAR missing, the convergence curve of the algorithm is as follows Figure 2 As shown in the figure, the corresponding root mean square error RMSE at the final iteration is 1.960.

Claims

1. A spatiotemporal data completion method based on composite non-convex functions and structural regression, characterized in that The following steps are involved: S1. storing the spatiotemporal data to be completed as a target tensor in the form of a three-dimensional tensor, wherein the spatiotemporal data to be completed is highway traffic speed information; S2, construct the WCT norm of the target tensor to capture the global multi-dimensional low-rank characteristics of spatiotemporal data in the transform domain; S3, constructing a structural regression model of the target tensor to reveal complex spatiotemporal data correlation patterns; S4. Combining the WCT norm and the structural regression model, a spatiotemporal data completion model is constructed, and auxiliary variables and additional constraints are introduced to transform the objective function of the spatiotemporal data completion model; S5. Use the alternating direction multiplier method to solve the transformed objective function, obtain the optimal solution of the target tensor, and complete the completion of the spatiotemporal data; In S1, the process of storing the spatiotemporal data to be completed as a target tensor in the form of a three-dimensional tensor is: S1.

1. Represent spatiotemporal data with missing data as a three-dimensional tensor , is the real data tensor, where express dimensional real vector space, n1 represents the number of devices used to collect data, n2 represents the number of time periods for collecting data on each natural day, and n3 represents the number of days for collecting data; S1.2, use express The index of the observation in , correspondingly, is express The index of missing data in; S1.

3. Constructing a three-dimensional tensor of spatiotemporal data to be repaired , X is the target tensor, The observations in are transferred to the corresponding positions of X; In S2, the WCT norm of the target tensor It is expressed as: （1）； In the formula, w u is a non-negative weight coefficient, u represents the index of the mode of the tensor, u=1, 2, 3, and w1+w2+w3=1; represents the tensor transformation of X on the u-th mode; for The lth slice matrix, the rest of the superscript Similarly, n u is the number of slice matrices; express The i-th singular value of , m is the number of singular values; is a composite non-convex function; In S3, the structural regression model is expressed as: （3）； In the formula, , , , They are X, P, , The expansion matrix on the second mode, the remaining subscripts Similarly; H is the autoregressive coefficient matrix; represents the Hadamard product; Z represents the sample coefficient matrix; represents the random error matrix; In the above S4, the method of constructing the spatiotemporal data completion model and transforming the objective function of the spatiotemporal data completion model is: S4.

1. The constructed spatiotemporal data completion model is expressed as: （5）； In the formula, , , is a non-negative parameter; the subscript F represents the Frobenius norm; the subscript 1 represents the L1 norm; The constraints are: （6）； In the formula, represents the diagonal matrix of Z, and the rest Similarly; S4.

2. Introducing auxiliary variables M1, M2, M3, D, G, U and V, formula (5) is transformed into the following objective function: （7）； In the formula, Indicates M u The tensor transformation on the u-th mode, M u That is, M1, M2, M3; the subscript WCT represents the WCT norm; the constraints of each auxiliary variable are: M u =X,D=X, , , ; The constraints for the target tensor X are: （8）； S4.3, the augmented Lagrangian function corresponding to equation (7) is expressed as: （9）； In the formula, y u , y4, Y5, Y6, and Y7 are all Lagrange multipliers; is the penalty parameter.

2. The spatiotemporal data completion method based on composite non-convex function and structural regression according to claim 1, characterized in that: In the S2, Specifically expressed as: （2）； Where x represents the input of the composite non-convex function; p is a non-negative parameter, 0<p≤1.

3. The spatiotemporal data completion method based on composite non-convex function and structural regression according to claim 1, characterized in that: H is represented by: （4）。 4. The spatiotemporal data completion method based on composite non-convex function and structural regression according to claim 1, characterized in that: In the above S5, the solution process is: S5.

1. Input non-negative weight coefficient w u , non-negative parameter , , , iteration stop parameter , and the maximum number of iterations ; S5.

2. Initialize parameters to get the initial value X of the target tensor 0 , the initial value of the sample coefficient matrix Z 0 , initial value of auxiliary variable G 0 、V 0 , U 0 , the initial value of the Lagrange multiplier , ,as well as , , ; S5.3, iteratively update each variable in S5.2 using the alternating direction multiplication method; S5.4, determine whether the iteration stop condition is met, if the iteration stop condition is met, stop the iteration, otherwise update k=k+1, k represents the number of iterations, and return to S5.3 to continue a new round of iteration; S5.

5. After the iteration stops, the final completed tensor is output, which is the target tensor X, to obtain the final completed spatiotemporal data.

5. The spatiotemporal data completion method based on composite non-convex function and structural regression according to claim 4, characterized in that: In S5.3, the process of iteratively updating the variables using the alternating direction multiplier method is: S5.3.

1. Update auxiliary variable M u , the process is as follows: S5.3.1.1, respectively and Do tensor singular value decomposition t-SVD, , , are respectively , X, y u , the superscript k indicates the kth iteration, and the other superscript k are similar; Indicated in Tensor transformation on the u-th mode; After decomposition, we get the corresponding and , where Q1, Q2, R1, R2 are orthogonal tensors; and are the conjugate transposes of R1 and R2 respectively; S1 and S2 are f-diagonal tensors; S5.3.1.

2. Calculation of the k+1th iteration , that is , the superscript k+1 indicates the k+1th iteration, and the other superscript k+1 is the same, calculation The formula is as follows: （10）； In the formula, is an f-diagonal tensor that satisfies: （11）； In the formula, Represents the element in S1 with index (a, b, j), 1≤a≤ , 1≤b≤ , 1≤j≤ , the rest are similar; is the threshold value, ; yes The derivative of is expressed as: （12）； Where x represents the input of the composite non-convex function; p is a non-negative parameter, 0<p≤1; S5.3.1.3, Yes Perform the inverse tensor transformation on the u-th mode to obtain M at the k+1th iteration u , that is ; S5.3.

2. Update the auxiliary variable D. The formula is as follows: （13）； In the formula, represents the inverse operation of the second mode expansion; S5.3.

3. Update the auxiliary variable G. The formula is as follows: （14）； In the formula, Represents element-wise division of two matrices; is a matrix whose elements are all 1; S5.3.

4. Update the auxiliary variable V. The formula is as follows: （15）； Where I is the identity matrix; the superscript T indicates transpose; S5.3.

5. Update the auxiliary variable U. The formula is as follows: （16）； Where A is an intermediate matrix, ; S5.3.

6. Update the auxiliary variable Z. The formula is as follows: （17）； In the formula, J is the intermediate variable, , is the intermediate variable, ; S5.3.

7. Update the target tensor X. The formula is as follows: （18）； In the formula, B is an intermediate variable, expressed as: （19）； S5.3.

8. Update the Lagrange multiplier as follows: （20）； In the formula, Indicates update.

6. The spatiotemporal data completion method based on composite non-convex function and structural regression according to claim 5, characterized in that: In S5.4, the iteration stopping condition is: when or , stop the iteration.

Images