A method based on truncation L 2,P A Scalable Industrial Time Series Data Completion Method Based on Norms

Abstract

This invention belongs to the field of industrial time-series data completion technology, specifically relating to a method based on truncated L... 2,P A scalable industrial time-series data completion method based on norms includes: acquiring multi-dimensional industrial time-series data through distributed sensors; processing missing values ​​in the acquired tensors to obtain sparse tensors to be completed, and constructing a low-rank tensor completion model; integrating linear unitary transforms to enhance the scalability of the model; and defining a truncation L... 2,P Norm, based on truncated L 2,P A low-order smooth tensor completion model based on the norm is proposed. This model decomposes a complex optimization problem into multiple parallelizable subproblems to obtain the objective solution. The objective solution is then adapted to an industrial time-series data format to obtain completed industrial time-series data. This invention utilizes truncated L... 2,P The norm better approximates the tensor rank, and by integrating linear unitary transformation, it can complete high-dimensional, large-scale industrial time series data, effectively improving data quality and providing reliable support for the accurate analysis and optimization of industrial processes.

Description

Technical Field

[0001] This invention belongs to the field of data completion technology, specifically relating to a method based on truncation L 2,P A scalable industrial time-series data completion method based on norms. Background Technology

[0002] Time series data is a dataset continuously recorded over time in an industrial production process. It typically encompasses multiple variables such as flow rate, temperature, pressure, and composition, and these variables are closely correlated. Industrial time series data, collected at continuous time intervals, can capture the dynamic changes in the production process. This data can be applied to production monitoring, fault diagnosis, quality control, energy management, and industrial intelligence, serving as a key tool for driving digital transformation of factories, improving efficiency, and promoting sustainable development. With the development of information technology, innovations in intelligent algorithms, data acquisition, sensing devices, and high-efficiency computing are driving its application value across various industries.

[0003] In actual production, missing industrial time-series data is common due to sensor hardware failures or poor acquisition conditions, easily leading to misleading conclusions and erroneous decisions. Industrial time-series data imputation aims to model based on known data and uncover inherent relationships to fill in missing values. Existing methods fall into three categories: First, single-time-series modeling, such as linear regression and autoregressive models, cannot characterize the correlation and dependence of different sensors in a spatiotemporal context, making it difficult to capture the dynamic changes and trends in industrial data; second, methods based on neural networks, but these are based on the principle of minimizing empirical risk and are prone to overfitting in small-sample scenarios, limiting their practical application; third, methods using tensor / matrix structures to model multivariate time-series data, however, the definition of tensor rank is not unique, and calculating tensor rank becomes a complex problem when dealing with high-dimensional, large-scale industrial time-series data tensors. Therefore, there is still significant room for improvement in achieving high-quality, efficient imputation of high-dimensional, large-scale industrial time-series data.

[0004] Many tensor completion methods capture global information by minimizing the tensor rank, but the definition of tensor rank is not uniform. CP rank calculation is an NP-hard problem, so convex or non-convex alternatives are often designed according to different definitions. Existing algorithms suffer from long running times (e.g., CP-WOPT) or poor recovery results (e.g., WTucker). Currently, Tucker rank is widely used, typically approximating the tensor rank by the sum of the kernel norms of the tensor expansion matrices in different directions. However, this approach loses crucial information, affecting subsequent tasks. Moreover, the kernel norm summation is a non-convex optimization, consuming significant resources and increasing complexity when handling large-scale tensors. Furthermore, to improve model scalability and enhance completion effects on large-scale data, linear transformations are introduced to decompose the large-scale tensor learning problem into a series of "smaller" subproblems for solving. For example, in large-scale image or video partial observation tensor learning, linear transformations such as Fourier transform and wavelet transform play a crucial role in constructing accurate and efficient multidimensional tensor completion models. Existing methods establish the correlation between tensor rank and tensor SVD, introducing frequency-domain invertible linear operators such as Fourier transform and cosine transform into tensor completion models. These low-rank tensor completion methods integrating invertible linear transforms significantly outperform methods based on weighted tensor kernel norms. The Discrete Fourier Transform is currently the most widely used invertible linear transform. To further improve the performance of low-rank tensor completion models, it is still necessary to explore suitable linear transforms. Summary of the Invention

[0005] To solve the above technical problems, the present invention provides a method based on truncated L 2,P Scalable industrial time-series data completion methods based on norms include:

[0006] S1: Collect multidimensional time series data through distributed sensors, and convert the three-dimensional collected data with missing values ​​into tensor form to obtain the observation tensor;

[0007] S2: Process missing values ​​of the observed tensor to form a sparse tensor to be completed, and establish a low-rank tensor completion model;

[0008] S3: Enhance model scalability by integrating linear unitary transformations;

[0009] S4: Define the truncation L 2,P The norm is substituted into the completion model, and combined with the minimization of quadratic variation, a low-tube-rank smooth tensor completion model is constructed.

[0010] S5: Solve using the alternating direction multiplier method with truncated L. 2,P The objective solution is obtained by smoothing and completing the low-rank tensor model of the norm.

[0011] S6: Adapt the target solution to an industrial time-series data format and complete the data completion.

[0012] The beneficial effects of this invention are:

[0013] This invention constructs the collected industrial time-series data into a tensor form, fully utilizing the correlation and dependency of different sensors in the spatiotemporal context to build a more accurate completion model. Furthermore, to further improve interpolation accuracy, a truncated L is introduced. 2,P Norms, by preserving the features of large singular values ​​and imposing a larger threshold penalty on small singular values ​​through singular value decomposition and shrinking threshold operators, better approximate tensor rank and avoid the problem of excessive singular value constraints. At the same time, by integrating linear unitary transformation, the large-scale tensor learning problem is decomposed into a series of "smaller" subproblems and solved, which significantly improves the relevance to high-dimensional data. Attached Figure Description

[0014] Figure 1 This invention is based on truncating L 2,P Flowchart of tensor completion method for industrial time series data with norm;

[0015] Figure 2 This is a unitary transformation diagram of the third-dimensional expansion of the third-order tensor according to an embodiment of the present invention;

[0016] Figure 3 This is a comparison chart of MAE (Magnitude of Effect) data used to complete data in a random missing pattern on an energy production monitoring dataset using different methods, according to an embodiment of the present invention.

[0017] Figure 4 This is a comparison chart of MAE (Data Envelope Equivalent) used to complete data in a random missing pattern on a power transformer dataset using different methods, according to an embodiment of the present invention. Detailed Implementation

[0018] 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. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] A method based on truncation L 2,P Scalable industrial time-series data completion methods based on norms, such as Figure 1 As shown, it includes the following steps:

[0020] S1. Collect multidimensional industrial time-series data through distributed sensors and convert the collected data with missing values ​​into observation tensors.

[0021] Specifically, in the acquisition of industrial time-series data, it is assumed that the sensor measures data once at regular time intervals.

[0022] S2. The missing values ​​of the observed tensor are processed to obtain the sparse tensor to be completed, and a low-rank tensor completion model is constructed.

[0023] Specifically, for missing data in the observation tensor, step S2 involves setting the missing values ​​to zero to generate a sparse tensor to be filled in.

[0024] Or more simply:

[0025] Specifically, missing value handling can be represented as:

[0026]

[0027] in, Ω is the target completion tensor, and Ω is the set of position indices of the observed elements. It is an observation tensor that transforms collected data with missing values. This is the projection function. `ij` is the index value, representing the observation tensor. The value of the element at the ij-th position.

[0028] Specifically, industrial time-series data exhibits certain similarities across different dimensions and possesses a degree of stability due to its acquisition over consecutive time intervals. Based on this, the observation tensor derived from industrial time-series data shows spatial correlation between its horizontal and lateral slices. Simultaneously, the frontal slices of the observation tensor exhibit temporal correlation. Given the high correlation among these slices, the observation tensor can be considered to possess approximately low-rank structural characteristics. Therefore, the data completion problem can be transformed into a low-rank tensor completion problem, where the low-rank tensor completion model using the nuclear norm as a convex substitute for the tensor rank is expressed as:

[0029]

[0030] in, It is to complete the tensor The k-th order tensor expansion matrix, where N is the tensor order. Tensor The nuclear norm of the modulo k expansion matrix.

[0031] S3. To improve the model's scalability and better fill in large-scale data, the model integrates linear unitary transformation.

[0032] Specifically, define Tensor The k-th mode expansion matrix is ​​transformed into the folding operator of the original tensor by the folding operator. k (·)

[0033] For any given tensor Its positive unitary transformation is defined as:

[0034]

[0035] Where Φ[·] is used to denote the unitary transformation, and Φ is the unitary matrix.

[0036] Correspondingly, the inverse unitary transform is defined as:

[0037]

[0038] Combining the forward unitary transform and the inverse unitary transform, we can obtain This invention selects tensors The third mode expansion matrix The left singular vector matrix of the singular value decomposition is taken as the unitary matrix Φ. The specific implementation of the forward and inverse unitary transformations is as follows: Figure 2 Place

[0039] S4. Define the truncation L 2,P The norm is then substituted into the complete model and combined with quadratic variation minimization to obtain the truncated L. 2,P Low-rank smooth tensor completion model of norm.

[0040] Specifically, when using the nuclear norm as an approximation of the tensor rank function, the nuclear norm uniformly accumulates singular values ​​during the solution process, failing to distinguish between singular values ​​of different magnitudes. Instead, it shrinks all singular values ​​to the same extent, resulting in the loss of useful information. To better approximate the tensor rank function, this invention uses a truncated L... 2,P Norm approximates the rank function.

[0041] Specifically, the L of the matrix 2,P Norm is defined as:

[0042]

[0043] in, Indicates taking L of the matrix 2,P Norm; the singular value decomposition of matrix X is U∑V T Then we have Σ=Diag(σ i (X)); σ i Let σ1 be the singular values ​​of the matrix, i.e., the nuclear norm of the matrix, and let the singular values ​​be ordered from largest to smallest: σ1 ≥ σ2 ≥ … ≥ σ3. min {m,n}≥0; p is the contraction parameter.

[0044] Due to the L of the matrix 2,P Norms cannot be directly applied to tensors; for example, L... 2,P Norm extends to tensor patterns, for any given L in tensor mode 2,P Norms are defined as follows

[0045]

[0046] Will Perform a unitary transformation along the third dimension n3. This represents the k-th forward slice after transformation, in matrix form. Indicates forward slice The i-th singular value after singular value decomposition.

[0047] Furthermore, we give the definition of the tensor truncation norm: for any third-order tensor Given a positive integer t = min(n1, n2), the truncated nuclear norm is in the form of the rank sum of the minimum singular values ​​of the tensor, i.e.

[0048]

[0049] In the formula: The truncated nuclear norm of the tensor. This represents the set of all forward slices after the transformation. Indicates forward slice The i-th singular value, Tensor The i-th singular value, with a truncation parameter r. <min(n1,n2)

[0050] Combining the two norm definitions of tensors, we give the tensor truncation L. 2,P Norm definition:

[0051]

[0052] in: Indicates truncation L 2,P Norm.

[0053] To solve the optimization problem, the tensor truncation L is given. 2,P The singular value threshold decomposition of the norm is defined as follows:

[0054] Arbitrary tensor Tensor Singular Value Decomposition (t-SVD) can be expressed as Based on this, given the threshold parameter μ, the tensor The singular value threshold is

[0055]

[0056] Define truncation L 2,P The shrinkage threshold operator is The following equation can be obtained:

[0057]

[0058] According to the t-SVD theorem, we have ∑ is the input tensor of the optimization problem, which can be the observation tensor with missing values, and ∑ is the singular value tensor. In representing optimization problems The solution, Generally, the tensor is completed. Here, μ is the threshold parameter for singular value thresholding, and p is the shrinkage parameter. When -∞ < p < 1, the smaller the singular value, the greater the threshold penalty; only the truncated singular values ​​are subjected to Lt. 2,P The form of contraction preserves the first r singular values. Furthermore, ||·|| F This represents the Frobenius norm. Regarding... have Therefore, it can be expressed as:

[0059]

[0060] Where ijk represents the index of a singular value in the singular value tensor; for any value Addition function (·) + Defined as: a + =max{a,0}.

[0061] Furthermore, L will be truncated. 2,P Substituting the norm into the low-rank tensor completion model and combining it with quadratic variation minimization, we obtain the result using truncated L. 2,P The low-tube-rank smooth tensor completion model of the norm is expressed as:

[0062]

[0063] in, This is an observation time series matrix containing missing values. M represents the number of sensors, I is the number of time points per day, and J is the time span of the data. Variables Z is the target completion tensor, and the variable Z is the time series matrix. t-1 and z t These are the (t-1)th and tth columns of matrix Z, respectively. The positive weight parameter λ is used to balance the truncation of L. 2,P Norms and quadratic variation terms maintain the low-rank structure of the data while ensuring the smoothness of the time series. It is a positive tensor quantization operator It can convert time-space sequence matrices into third-order tensors. Conversely, it can be converted using its inverse operator. Convert the tensor into the original matrix.

[0064] S5. Based on the use of truncation L 2,P The low-rank smooth tensor completion model of the norm is solved by the alternating direction multiplier method to obtain the objective solution.

[0065] Specifically, in step S5, the alternating direction multiplier method is used to solve for the truncated L. 2,P The low-tube-rank smooth tensor completion model of the norm yields the objective solution, including:

[0066] S51. Calculate the updated usage truncation L 2,P Augmented Lagrangian function of low-tube-rank smooth tensor completion model of norm;

[0067] S52. Apply the Alternating Directional Multiplier Method (ADMM) to the new truncation L. 2,P The low-tube rank smooth tensor completion model of the norm is solved iteratively, including:

[0068] S521. First, the new use of truncation L is given. 2,P Augmented Lagrangian function of the norm-based low-tube-rank smooth tensor completion model:

[0069]

[0070] Where <, > are the inner product. It is a Lagrange multiplier, and ρ is the constraint penalty parameter.

[0071] S522. Adopt ADMM to truncate the new L. 2,P The iterative solution problem of the norm-based low-rank smooth tensor completion model is transformed into the following three sub-problems:

[0072]

[0073] Among them, superscript Indicates the iteration round, for example For tensor No. The result updated in the next iteration.

[0074] S523. For the three sub-problems (15)-(17) above, iteratively solve them alternately to obtain the target solution, wherein the update order is as follows:

[0075] Specifically, in the first In the next iteration, the alternating solution includes:

[0076] a) Update

[0077] Subproblem (15) expands to:

[0078]

[0079] make but:

[0080]

[0081] right Perform a forward unitary transformation:

[0082]

[0083] in, It is a tensor The matrix expanded along the third mode, selected The left singular vector matrix is ​​taken as the unitary matrix Φ.

[0084] Let the transformation matrix be... It is a unitary transformation tensor The singular value decomposition of the j-th forward slice is:

[0085]

[0086] Among them, U Φ Let Σ be a left singular vector matrix. Φ It is a singular value matrix. Let be the transpose of the right singular vector matrix. From the tensor singular value thresholding theory, the optimal solution to equation (19) is:

[0087]

[0088] In the formula, It is to complete the tensor The estimated data for the j-th positive slice, i.e., the j-th day, is obtained through... Tensor solutions can be converted into matrix solutions.

[0089] b) Update Z:

[0090] Subproblem (16) can be expanded into the following expression:

[0091]

[0092] Z is a matrix consisting of M time series containing T (T = IJ) consecutive time points. Its second variation It can be represented as:

[0093]

[0094] in, 0 T-1 It is a zero vector of length T-1, I T-1 It is a T-1 order identity matrix. Let Ψ = Ψ2 - Ψ1, equation (23) can be rewritten as follows:

[0095]

[0096] Here, tr(·) represents the trace of the matrix.

[0097] make have to:

[0098]

[0099] Then, the iteration can be obtained. The least squares solution is:

[0100]

[0101] c) Update

[0102]

[0103] d) Update penalty parameters:

[0104]

[0105] Where 1.05 is the learning rate of the penalty parameter, ρ max This indicates the maximum value of the penalty parameter.

[0106] S5. Convert the target solution into the format of industrial time series data to obtain the completed industrial time series data.

[0107] In one embodiment, to verify the effectiveness of the method of the present invention, experiments were conducted using the present method and several existing tensor completion methods. The data used in the experiments were the energy production monitoring dataset on nrel and the power transformer dataset on github.

[0108] The energy production monitoring dataset uses solar power generation data from Alabama in 2006, sampled every 10 minutes from 137 power generation sites. A test set was constructed based on 100 days of data, ultimately forming a 137×144×100 three-dimensional tensor structure.

[0109] The power transformer dataset was collected from a region in China. The dataset records a total of 7 data items with a sampling step of 15 minutes. It is sampled 4 times per hour and 96 times per day. Using 725 days of data from this dataset for testing, it can be constructed into a tensor dataset of size 7×96×725.

[0110] Mean absolute error was selected as the evaluation index.

[0111]

[0112] Where y and These represent the actual test data and the completed data, respectively. When calculating the MAE, only the reconstruction error of the missing data portion is considered. In the test cases, the test data is constructed as a third-order tensor because, compared to higher-dimensional tensors, third-order tensors are not only more visually appealing but also easier to interpret. In the different examples shown, Figure 3 The MAE results at different missing rates are presented when this method is used to complete energy production monitoring data; while Figure 4 The figure shows the MAE performance after performing the same processing on power transformer data. As can be seen from the figure, the overall performance of the proposed method for data completion is superior to other tensor completion methods.

[0113] The data completion method proposed in this invention demonstrates significant advantages on both types of industrial datasets. For example... Figure 3 and Figure 4 As shown, under the same missing rate, this method consistently maintains a lower MAE value compared to other completion algorithms. As the missing rate gradually increases from 10% to 90%, the error of this method only shows a slow upward trend, and its advantage continues to expand, while the error of the comparison methods increases significantly. This difference is particularly evident when the missing rate exceeds 70%, fully validating the robustness and stability of the proposed method under high missing rate conditions.

[0114] Compared with existing technologies, the MAE of completed industrial time-series data is generally lower than that of other comparative methods. The method of this invention obtains the unitary transformed tensor to be completed by performing a unitary transformation on the matrix expanded along the third mode of the tensor to be completed. The unitary transformed tensor is divided into slice matrices according to the time span (corresponding to the third mode). Each slice matrix is ​​completed, and a truncation L is applied when the slice matrix undergoes singular value thresholding. 2,P The method employs norm-based contraction and considers the smoothness of the time series under the constraint of minimizing quadratic variation, enhancing the stability of the data completion. The completed slice matrix is ​​then reconstructed into a complete data tensor through inverse unitary transformation and folding operations. This invention further strengthens the effectiveness of the industrial time series data completion method for large-scale data, achieving high accuracy even with high data missing rates.

[0115] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A scalable industrial time series data imputation method based on truncated L 2,P norm, characterized in that, The method comprises the following steps: S1: collecting multi-dimensional time series data through a distributed sensor, and converting three-dimensional collected data with missing values into a tensor form to obtain an observation tensor; S2: processing missing values of the observation tensor to form a sparse tensor to be completed, and establishing a low-rank tensor completion model; The low-rank tensor completion model comprises: S21: constructing an initial low-rank tensor completion model according to the tensor to be completed; wherein, denotes a rank function, denotes an observed tensor with missing values, denotes a target completed tensor, denotes a set of known elements, denotes a completed tensor an element at a known element index; S22: using a kernel norm as a convex substitute for a rank function to obtain a final low-rank tensor completion model; where N denotes the tensor order, denotes the matrix kernel norm; S3: enhancing model scalability by integrating a linear unitary transformation; S4: define the cut-off L 2,P The norm is substituted into the completion model, combined with the quadratic variation minimization, and a low tube rank smoothing tensor completion model is constructed. The truncated L 2,P norm is substituted into a low-rank tensor completion model, combined with a quadratic variation minimization, to obtain a low-rank tensor completion model using the truncated L 2,P norm, including: wherein, denotes the observed time series matrix containing missing values, denotes the number of sensors, denotes the number of time points per day, denotes the time span of the data, denotes the time series matrix, z t−1 and z t denote the t−1th and tth column of the time series matrix denotes the positive weight parameter for balancing the L 2,P norm with the quadratic variation term, denotes the forward tensorization operator, denotes the set of position indices of the observed data, denotes the matrix the element at the index of the observed value, denotes the observed value;​ S5: solve the target solution by using the alternating direction multiplier method using the truncated L 2,P norm low-tubular-rank tensor smoothing completion model S6: adapting a target solution to an industrial time series data format to complete data completion.

2. A scalable industrial time-series data completion method based on truncated L 2,P An extensible industrial time-series data completion method based on norm, characterized in that, The step of processing missing values of the observation tensor to form a sparse tensor to be completed comprises: wherein, denotes the target completion tensor, denotes the observed tensor with missing values, denotes the projection function, denotes the index value, denotes the set of position indices of the observed elements, denotes the set of known elements, denotes the completion tensor the element at the known element index.

3. A truncated L 2,P An extensible industrial time-series data completion method based on norm, characterized in that, The step of enhancing model scalability by integrating a linear unitary transformation comprises: Definitions denotes the k-th mode unfolding matrix of the tensor , the folding operator of the original tensor into the k-th mode unfolding matrix is ; For any given tensor its forward unitary transform is defined as: its backward unitary transform is defined as: wherein denotes a unitary transform, denotes a unitary matrix, n1, n2, n3 correspond to the first, second, third dimension of the tensor, denotes the set of real numbers; Combining the forward unitary transformation and the reverse unitary transformation, we can get .

4. A scalable industrial time-series data completion method based on truncated L 2,P An extensible industrial time-series data completion method based on norm, characterized in that, Definition of the cut-off L 2,P Norms, including: S41: Definition of L2, P-norm: Given a matrix , for any : wherein, denotes the L2, P-norm of a matrix; it is assumed that the matrix has a singular value decomposition , then ; denotes the singular values of a matrix, and the singular values are ordered from large to small ; is a shrinkage parameter, denotes the set of real numbers, U denotes a left singular vector matrix, denotes the transpose of a right singular vector matrix, denotes a diagonalization operation, m, n denote two dimensions of a matrix , respectively; S42: L 2,P The norm is extended to the tensor mode, for a given The L2, P norm in the tensor mode is defined as follows: wherein n1, n2, n3 correspond to the first, second, third dimension of the tensor respectively, denotes the tensor transformed to a matrix form forward slice, denotes the forward slice singular value decomposition singular value; S43: the definition of a tensor truncated kernel norm is as follows: Given a third-order tensor and a positive integer = min , the truncated nuclear norm is in the form of the sum of the ranks of the smallest singular values of the tensor wherein, denotes the truncated nuclear norm of a tensor, denotes the set of all forward slices after transformation, denotes a forward slice of the tensor, denotes the th singular value of the tensor denotes the th singular value of the tensor, and has a truncation parameter <min ; S44: Get the truncated L of the tensor by combining the norm definition of the two tensors 2,P Norm definition: wherein denotes the truncation L of a tensor 2,P norm S45: Truncation L of a tensor 2,P The singular value threshold decomposition definition of a norm: for a tensor The tensor singular value decomposition (t-SVD) can be expressed as where denotes the left singular tensor, denotes the singular value tensor, denotes the transpose of the right singular tensor; based on this, the singular value threshold of a tensor is defined as follows, given a threshold parameter ​ wherein denotes the result of applying a threshold to the singular value tensor; The definition of the cut-off L 2,P The shrinkage threshold operator is The following equation can be obtained: According to the t-SVD theorem, we have , It is the input tensor of the optimization problem, which can be an observation tensor with missing values. It is a singular value tensor. In representing optimization problems The solution, where, The threshold parameter for singular value threshold decomposition. For contraction parameters; when The smaller the singular value, the greater the threshold penalty. Only the truncated singular values ​​are subject to L... 2,P Formal contraction, before One singular value is preserved, in addition. Denotes the Frobenius norm, regarding ,have Therefore, it can be expressed as: wherein denotes the position index of singular values in the singular value tensor, for any value addition function is defined as: .

5. A scalable industrial time-series data completion method based on truncated L 2,P An extensible industrial time-series data completion method based on norm, characterized in that, The alternating direction method of multipliers is used to solve the low-tubular-rank tensor completion model with the truncated L 2,P norm, which includes the following steps: S51. Calculate the updated usage cutoff L 2,P Augmented Lagrangian function of the low tube rank smoothing tensor completion model S52. Based on the augmented Lagrange function, the model solving problem is converted into three sub-problems for iterative and alternative solving.

6. A truncated L 2,P An extensible industrial time-series data completion method based on norm, characterized in that, Compute the updated usage cutoff L 2,P The augmented Lagrangian function of the low-tubular-rank smoothing tensor completion model, including: wherein, denotes an inner product, denotes a Lagrange multiplier, denotes a constraint penalty parameter, denotes a forward tensorization operator, denotes a target completion tensor, denotes a time series matrix, denotes a balancing truncation L 2,P a positive weight parameter for the quadratic variation term, z t−1 and z t denote the t−1th column and the tth column of the time series matrix X, respectively.

7. A truncated L 2,P The scalable industrial time-series data completion method based on norm, characterized in that, The step of converting the model solving problem into three sub-problems based on the augmented Lagrange function comprises: wherein, denotes an iteration round, denotes a target completion tensor, denotes a time series matrix, denotes a Lagrange multiplier, denotes a constraint penalty parameter, denotes a tensorization operator, denotes a Lagrangian function.

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