A low-rank single-sensor building energy consumption data repairing method and device

Abstract

The application provides a low-rank single-sensor building energy consumption data repairing method and device, and relates to the technical field of data repairing processing.The application reconstructs a one-dimensional energy consumption sequence into a three-dimensional tensor of hours, days and weeks, constructs a non-convex low-rank constraint term based on SCAD penalty to avoid excessive shrinkage of singular values, and fuses a time autoregressive regularization term and a week-scale periodicity regularization term to depict short-range autocorrelation and long-range periodic patterns, finally solves a unified optimization model through an alternating direction multiplier method, and introduces auxiliary variables corresponding to each regularization term until a preset stop condition is met, so that complete building energy consumption data is obtained after repairing is completed.The application can effectively improve the repairing accuracy and robustness of long-continuous missing building energy consumption data in a single-sensor scene.

Description

Technical Field

[0001] This invention relates to the field of building energy management and data processing technology, and more specifically, to a method and apparatus for repairing low-rank single-sensor building energy consumption data. Background Technology

[0002] In actual building energy consumption monitoring and data collection, data gaps often occur due to factors such as equipment failure, communication interruptions, or external disturbances. This not only disrupts the regularity of energy consumption sequences but also weakens the inference performance of subsequent data-driven models in energy consumption prediction and energy assessment. Therefore, accurately restoring missing energy consumption data is a crucial prerequisite for ensuring reliable building energy analysis and management.

[0003] Current research on building energy consumption data restoration largely relies on cross-variable correlations, using auxiliary information such as meteorological variables, building attributes, or other sensor measurements for multivariate restoration. However, when the target building lacks external covariates or multiple related variables are simultaneously missing, such multivariate restoration methods often fail, making the restoration of single-sensor building energy consumption sequences a research challenge. For single-sensor sequences, existing techniques have attempted to utilize the periodicity of the data, such as proposing restoration frameworks based on K-singular value decomposition or mixed factor analysis. However, these methods often over-rely on the correlation of periodic patterns, easily ignoring other characteristics of building energy consumption data besides periodicity. In addition, some studies stack single-sensor energy consumption sequences into tensors and use low-rank tensor completion techniques for restoration. Although Tucker decomposition or CP decomposition can handle nested seasonal structures to some extent, they often involve complex rank selection problems and have high computational costs. While tensor completion methods based on nuclear norm minimization can adaptively determine weights, the consistent shrinkage of the nuclear norm with singular values ​​easily leads to an underestimation of the actual rank, thus limiting further improvement in restoration accuracy under high missing values.

[0004] While non-convex penalty terms such as Smooth Cut-Off Absolute Bias (SCAD) demonstrate advantages in suppressing small singular values ​​and preserving dominant singular values ​​in multi-sensor traffic data restoration, effectively mitigating the shortcomings caused by nuclear norm uniformity contraction, corresponding technical solutions are still lacking in the field of single-sensor building energy consumption restoration. More importantly, if the restoration process relies solely on global low-rank constraints, it often fails to fully exploit the complex temporal features beyond the low-rank structure. Single-sensor building energy consumption sequences exhibit significant short-range autocorrelation and periodic similarity on a weekly scale. If these temporal structural features cannot be explicitly characterized in the restoration model, the restoration results will lack stability and expressiveness in scenarios with long continuous missing data.

[0005] In view of the above, this application is hereby submitted. Summary of the Invention

[0006] The present invention aims to provide a method and apparatus for repairing low-rank single-sensor building energy consumption data, in order to solve the defects of existing methods such as low repair accuracy, large rank estimation bias, and insufficient utilization of time structure features caused by long continuous missing data of building energy consumption monitoring data due to equipment failure or communication link interruption.

[0007] To solve the above-mentioned technical problems, the present invention is achieved through the following technical solution:

[0008] A method for repairing low-rank single-sensor building energy consumption data includes:

[0009] S1 collects a one-dimensional time series of building energy consumption from a single sensor, and performs three-dimensional tensor reconstruction according to hourly, daily, and weekly scales to obtain the three-dimensional observation tensor;

[0010] S2, Apply the smooth truncation absolute deviation penalty function to the matrix obtained by expanding the three-dimensional observation tensor along each mode to construct a low-rank constraint term;

[0011] S3, construct an autoregressive model based on the third mode expansion matrix of the three-dimensional observation tensor along the circumferential mode expansion, and construct a time autoregressive regularization term based on the autoregressive fitting residual;

[0012] S4. Based on the third modality expansion matrix, introduce the self-representation coefficient matrix and elastic network constraints to construct a periodic-scale periodic similarity regularization term;

[0013] S5, integrate the low-rank constraint term, time autoregressive regularization term and weekly-scale periodic similarity regularization term into a unified optimization objective function, and apply observation data consistency constraints;

[0014] S6. The unified optimization objective function is solved iteratively using the alternating direction multiplier method, and auxiliary variables are introduced to correspond to each regularization term until the preset stopping condition is met, so as to obtain the complete building energy consumption data after the repair is completed.

[0015] Preferably, S2 specifically comprises:

[0016] First, the three-dimensional observation tensor is expanded into matrices along three modes, which correspond to the hourly, daily, and weekly scales, respectively.

[0017] Singular value decomposition is performed on the expanded matrix of each mode to obtain the singular values ​​of each matrix;

[0018] The Smooth Cut-Off Absolute Bias Penalty Function is applied to each singular value, and then combined with the weights of each mode and the projection constraints of the observed elements to construct a low-rank constraint term with SCAD penalty; the expression of the low-rank constraint term is:

[0019] ;

[0020] ;

[0021] ;

[0022] in, This indicates finding the minimum value; The three-dimensional low-rank tensor to be completed; The three-dimensional observation tensor; Assign a modal index number; express Along the first The matrix of modal expansion; For the first The weights corresponding to modal expansion; For the first The SCAD norm of the modal expansion matrix; Indicates constraints; For projection operators, preserve the tensor Set the observed element corresponding to the index to zero, and set all other elements to zero; It is the set of indices of the observed elements in the three-dimensional observation tensor; , These are the regularization parameters and adjustment parameters for SCAD penalty, respectively. for The One singular value; for The smallest dimension; For SCAD penalty function; These are the variables of the SCAD penalty function, i.e., singular values.

[0023] Preferably, the expression for the time autoregressive regularization term is:

[0024] ;

[0025] in, This is the third mode expansion matrix; For time-regressive regularization; The order of the autoregressive model; , , These represent the hourly, daily, and weekly dimensions of the three-dimensional tensor, respectively. For the first The first line corresponding to the One autoregressive coefficient; The third mode matrix Line number Elements at each point in time.

[0026] Preferably, the expression for the periodic-scale periodicity regularization term is:

[0027] ;

[0028] in, This is the third mode expansion matrix; For periodic similarity regularization at the weekly scale; This is a self-representation coefficient matrix used to characterize the similarity between different weeks; It is the Frobenius norm; It is an L1 norm; Here is the regularization strength parameter for elastic networks, used to represent the regularization strength of elastic networks. Elastic network penalty items; The mixing coefficient of the elastic network; The diagonal elements of the self-representation coefficient matrix are 0; Indicates constraints.

[0029] Preferably, the expression for the unified optimization objective function is:

[0030] ;

[0031] in, This indicates finding the minimum value; Let be the three-dimensional low-rank tensor to be completed; A is the autoregressive coefficient matrix. It is a self-representation coefficient matrix; Assign a modal index number; For the first The weights corresponding to modal expansion; Representing a three-dimensional low-rank tensor Along the first The matrix of modal expansion; For the first The SCAD norm of the modal expansion matrix; The time-regression regularization strength parameter; For time-regressive regularization; The periodic similarity regularization intensity parameter is used for the periodic scale. For periodic similarity regularization at the weekly scale; Indicates constraints; For projection operators, preserve the tensor Set the observed element corresponding to the index to zero, and set all other elements to zero; It is the set of indices of the observed elements in the three-dimensional observation tensor; The three-dimensional observation tensor; This indicates that the diagonal elements of the self-representation coefficient matrix are 0.

[0032] Preferably, S6 specifically includes:

[0033] First, auxiliary tensors are introduced to correspond to each regularization term, transforming the multidimensional coupled optimization problem of the unified optimization objective function into an equivalent splitting form to reduce the coupling complexity between variables. The expression is as follows:

[0034] ;

[0035] in, These are the corresponding low-rank constraint terms with SCAD penalties. Matrix expanded along modes 1, 2, and 3 , , Auxiliary variables; The three-dimensional low-rank tensor to be completed; For the time autoregressive regularization term, the corresponding Auxiliary tensor; For the corresponding self-representation coefficient matrix Auxiliary variables used to handle constraints ; For the periodic similarity regularization term on a weekly scale, the corresponding Auxiliary variables; , , They are respectively , , The weights; In the periodic similarity regularization term at the week scale Auxiliary variables; It is the Frobenius norm; It is an L1 norm; Here is the regularization strength parameter for elastic networks, used to represent the regularization strength of elastic networks. Elastic network penalty items; The mixing coefficient of the elastic network; The diagonal elements of the self-representation coefficient matrix are 0; Indicates constraints; Representing auxiliary tensor Along the first The expansion matrix of the modes;

[0036] By introducing Lagrange multipliers and penalty parameters, the equality constraints are transformed into augmented Lagrange form, and the augmented Lagrange function is constructed, with the expression:

[0037] ；

[0038] in, To augment the Lagrange function; For Lagrange multipliers, ; For penalty parameters, , used to control the penalty strength of constraint terms in the augmented Lagrange function; Indicates the inner product;

[0039] Next, following the block descent strategy, during the iteration process, the update subproblems corresponding to each variable are solved sequentially, as expressed in the following expression:

[0040]

[0041] in, as auxiliary variables The The value is updated in the next iteration; Indicates the number of iterations; for Along the first The matrix of modal expansion Auxiliary variables; Represents the minimization operator; The corresponding variables are the first The value is updated in the next iteration; Indicates fixed division All variables outside of, for Perform a minimization solution;

[0042] When updating the auxiliary tensor, a closed update formula is obtained through singular value decomposition and SCAD shrinkage operator to efficiently realize low-rank constraints.

[0043] Among them, auxiliary tensor The update formula is:

[0044] ;

[0045] It can be equivalently rewritten using the completing the square method as follows:

[0046] ;

[0047] in, For nuclear norm; For penalty parameters; For the first In the next iteration, the corresponding Constrained Lagrange multipliers;

[0048] make The singular value decomposition is represented as:

[0049] ;

[0050] in, , The first The left and right orthogonal matrices of the singular value decomposition in the next iteration; For the first One singular value; For matrix The smallest dimension; It is the transpose symbol;

[0051] The closed-form update formula for the auxiliary tensor obtained by applying the SCAD contraction operator to the singular values ​​is as follows:

[0052] ;

[0053] ;

[0054] in, For SCAD shrinkage operators; For modal folding operators;

[0055] When the iteration meets the preset stopping condition, the iteration stops and the repaired 3D building energy consumption tensor is output.

[0056] Preferably, the preset stopping condition includes the residual being less than a preset threshold or reaching the maximum number of iterations.

[0057] Preferably, the autoregressive coefficient matrix A is estimated by minimizing the sum of squared predicted residuals of all weekly sequences, and the coefficients of each weekly sequence are updated in parallel using the least squares method during the iterative process of the alternating direction multiplier method.

[0058] Preferably, the Lagrange multiplier The update is based on the residual between the current repair tensor and the corresponding auxiliary variable, and is performed through gradient ascent.

[0059] The present invention also provides a device for repairing low-rank single-sensor building energy consumption data, comprising:

[0060] The three-dimensional tensor reconstruction unit is used to collect one-dimensional time series of building energy consumption from a single sensor and perform three-dimensional tensor reconstruction according to hourly, daily and weekly scales to obtain three-dimensional observation tensors.

[0061] A low-rank constraint term construction unit is used to apply a smooth truncation absolute deviation penalty function to the matrix obtained by expanding the three-dimensional observation tensor along each mode to construct low-rank constraint terms;

[0062] The time autoregressive regularization term construction unit is used to construct an autoregressive model based on the third mode expansion matrix of the three-dimensional observation tensor along the circumferential mode expansion, and to construct a time autoregressive regularization term based on the autoregressive fitting residuals.

[0063] The weekly-scale periodic similarity regularization term construction unit is used to construct a weekly-scale periodic similarity regularization term based on the third mode expansion matrix, by introducing a self-representation coefficient matrix and elastic network constraints.

[0064] An optimization objective function unit is used to integrate the low-rank constraint term, the time autoregressive regularization term, and the periodic-scale periodic similarity regularization term into a unified optimization objective function, and to impose observation data consistency constraints.

[0065] The iterative solution unit is used to iteratively solve the unified optimization objective function using the alternating direction multiplier method, and introduces auxiliary variables corresponding to each regularization term until the preset stopping condition is met, so as to obtain the complete building energy consumption data after the repair is completed.

[0066] The present invention also provides a device for repairing low-rank single-sensor building energy consumption data, including a processor and a memory. The memory stores a computer program that can be executed by the processor to realize the method for repairing low-rank single-sensor building energy consumption data as described above.

[0067] The present invention also provides a computer-readable storage medium storing computer-readable instructions, which, when executed by a processor of the device on which the computer-readable storage medium is located, implement a method for repairing low-rank single-sensor building energy consumption data as described above.

[0068] In summary, compared with the prior art, the present invention has the following beneficial effects:

[0069] First, this invention effectively solves the problem of rank underestimation caused by excessive contraction of large singular values ​​in traditional nuclear norms by introducing a smooth truncation absolute deviation non-convex penalty term. In the process of building energy consumption data restoration, this technical solution can more completely preserve the main components of the energy consumption sequence, making the restored data closer to the true value in terms of overall energy distribution and peak characteristics. Experiments show that this invention significantly reduces the relative error of building energy consumption data restoration and improves data fidelity.

[0070] Second, this invention constructs a multi-scale feature mining mechanism by fusing time autoregression and periodic similarity regularization. The time autoregression term utilizes the short-range evolution patterns of energy consumption data to provide interpolation support based on historical trends for long, continuous missing regions; the periodic similarity term utilizes the long-term stability of building operation patterns, compensating for the information deficiency caused by the lack of external covariates in single sensors through the integration of cross-weekly information. This synergistic effect of multi-dimensional constraints makes this invention exhibit extremely strong robustness when dealing with high missing rates and complex missing patterns.

[0071] Third, this invention employs an elastic network self-representation mechanism to characterize periodic similarity, overcoming the limitations of single sparse constraints in processing strongly correlated samples. In building energy consumption scenarios, there is a high correlation between adjacent weeks or weeks with similar weather conditions. The grouping effect generated by the elastic network enables the synchronous extraction of features from these correlated weeks, thereby improving the learning accuracy of the self-representation coefficients and providing a more reliable structural prior for the accurate recovery of missing data.

[0072] Fourth, the solution framework based on the alternating direction multiplier method in this invention exhibits good convergence and computational efficiency. By employing variable splitting techniques, the complex non-convex optimization problem is transformed into subproblems with closed-form solutions or easily solvable subproblems, reducing the difficulty of algorithm implementation. Simultaneously, the decoupled structure of each subproblem enables parallel computation, meeting the data repair needs of large-scale building energy consumption monitoring systems.

[0073] In summary, this invention achieves high-precision and robust repair of single-sensor building energy consumption data, providing high-quality, continuous, and complete data support for subsequent building energy conservation diagnosis, energy consumption prediction, and demand-side response, and has significant engineering application value. Attached Figure Description

[0074] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained from these drawings without creative effort.

[0075] Figure 1 This is a flowchart illustrating a method for repairing low-rank single-sensor building energy consumption data, as provided in Example 1.

[0076] Figure 2 This is a schematic diagram of reconstructing a one-dimensional time series of univariate building energy consumption into a three-dimensional tensor of hours, days, and weeks, as provided in Example 1.

[0077] Figure 3 The diagram shows the matrix expansion of the three-dimensional tensor provided in Example 1 along different modes, where Figure (a) shows the expansion effect of the first mode, Figure (b) shows the expansion effect of the second mode, and Figure (c) shows the expansion effect of the third mode.

[0078] Figure 4 This is a schematic diagram illustrating the principle of applying a d-order autoregressive model to weekly energy consumption data, as provided in Example 1.

[0079] Figure 5 This is a schematic diagram illustrating the principle of characterizing periodic similarity on a week-scale based on a self-representation mechanism, as provided in Example 1.

[0080] Figure 6 The original time series diagram of hourly electricity demand used in the experiment provided for Example 1.

[0081] Figure 7 The figures provided in Example 1 show a comparison of the repair results of the method of the present invention and the traditional nuclear norm repair method. Figure (a) shows the repair result based on the nuclear norm, and Figure (b) shows the repair result of the method of the present invention.

[0082] Figure 8 This is a schematic diagram of a device for repairing low-rank single-sensor building energy consumption data provided in Embodiment 2.

[0083] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. Detailed Implementation

[0084] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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 a part of the embodiments of the present invention, not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0085] Example 1

[0086] Embodiment 1 of the present invention provides a method for repairing low-rank single-sensor building energy consumption data, which can be implemented by a low-rank single-sensor building energy consumption data repair device (hereinafter referred to as repair device), specifically, executed by one or more processors within the repair device.

[0087] In this embodiment, the repair device may be an electronic device equipped with a processor, the processor having a computer program for the repair method of low-rank single-sensor building energy consumption data and the computer program being executable, such as a computer, smartphone, smart tablet, workstation, etc., without limitation.

[0088] like Figure 1As shown, a method for repairing low-rank single-sensor building energy consumption data is proposed. This method addresses the problem of missing single-sensor building energy consumption data and achieves high-precision repair based on three-dimensional tensor modeling, multiple regular constraints, and alternating direction multiplier method (ADMM) optimization. The method includes steps S1 to S6.

[0089] S1 collects a one-dimensional time series of building energy consumption from a single sensor, and performs three-dimensional tensor reconstruction according to hourly, daily, and weekly scales to obtain the three-dimensional observation tensor.

[0090] Building energy consumption time series typically exhibit multi-layered nested periodicity (e.g., patterns at hourly, daily, and weekly scales). To fully explore the characteristics of building energy consumption data, the univariate series is reconstructed into a three-dimensional tensor with dimensions of "hour × day × week".

[0091] This step breaks through the limitations of one-dimensional time-series data, transforming the raw one-dimensional time-series data of building energy consumption collected by a single sensor. For example... Figure 2 As shown, one-dimensional time series data is reconstructed into three-dimensional observation tensors according to three time dimensions: hourly, daily, and weekly. This completes the multi-scale structural transformation of time series data, providing a data foundation for subsequent mining of low-rank, periodic, and time series patterns. It realizes the multi-scale structuring of single-sensor data without relying on multiple sensors / external covariates, and is suitable for single-sensor monitoring scenarios.

[0092] S2, apply the smooth truncation absolute deviation penalty function to the matrix obtained by expanding the three-dimensional observation tensor along each mode to construct a low-rank constraint term.

[0093] Specifically, such as Figure 3 As shown, the three-dimensional observation tensor is first expanded into matrices along the three modes to obtain the expansion matrix corresponding to each mode; the three modes correspond to the hourly scale, daily scale and weekly scale respectively.

[0094] Singular value decomposition is performed on the expanded matrix of each mode to obtain the singular values ​​of each matrix;

[0095] The Smooth Cut-Off Absolute Deviation (SCAD) penalty function is applied to each singular value, and then combined with the weights of each mode and the projection constraints of the observed elements to construct a low-rank constraint term with SCAD penalty, which replaces the traditional nuclear norm to achieve low-rank constraints.

[0096] The expression for the low-rank constraint term is:

[0097] ;

[0098] ;

[0099] ;

[0100] in, This indicates finding the minimum value; The three-dimensional low-rank tensor to be completed; The three-dimensional observation tensor; Assign a modal index number; express Along the first The matrix of modal expansion; For the first The weights corresponding to modal expansion; For the first The SCAD norm of the modal expansion matrix; Indicates constraints; For projection operators, preserve the tensor Set the observed element corresponding to the index to zero, and set all other elements to zero; It is the set of indices of the observed elements in the three-dimensional observation tensor; , These are the regularization parameters and adjustment parameters for SCAD penalty, respectively. for The One singular value; for The smallest dimension; For SCAD penalty function; These are the variables of the SCAD penalty function, i.e., singular values.

[0101] SCAD penalty is widely considered an effective non-convex penalty function in statistical learning, possessing the properties of inducing sparsity and preserving continuity, and its proximal operator has a closed-form expression. Introducing SCAD into low-rank tensor completion can effectively suppress smaller singular values ​​while preserving dominant singular values ​​as much as possible, thereby improving completion accuracy.

[0102] This step uses SCAD non-convex penalty to replace the traditional nuclear norm, achieving more accurate tensor low-rank constraints. It solves the defects of traditional nuclear norms such as excessive shrinkage of large singular values ​​and underestimation of rank, and fully preserves the main components, overall energy distribution and peak characteristics of the energy consumption sequence, significantly reducing the relative error of repair and improving data fidelity.

[0103] S3, construct an autoregressive model based on the third mode expansion matrix of the three-dimensional observation tensor along the circumferential mode expansion, and construct a time autoregressive regularization term based on the autoregressive fitting residuals.

[0104] To more accurately characterize the local temporal dependence in building energy consumption data, this invention introduces a temporal autoregressive (AR) regularization term. In this term, time... The observed values ​​were from the previous time step Prediction is made by linear combination of observed values, with a time autoregressive regularization term AR( The modeling process is as follows: Figure 4 As shown.

[0105] In this step, the third mode expansion matrix of the three-dimensional observation tensor along the circumferential mode expansion is extracted. Using this matrix as the modeling object, a time autoregressive model is constructed to explore the short-range time evolution law of energy consumption data, calculate the fitting residual of the time autoregressive model, and construct a time autoregressive regularization term based on the residual.

[0106] The expression for the time autoregressive regularization term is:

[0107] ;

[0108] in, This is the third mode expansion matrix; For time-regressive regularization; The order of the autoregressive model; , , These represent the hourly, daily, and weekly dimensions of the three-dimensional tensor, respectively. For the first The first line corresponding to the One autoregressive coefficient; The third mode matrix Line number Elements at each point in time.

[0109] This step involves fitting an autoregressive model to each weekly time series and penalizing its residual error, thereby enhancing local temporal smoothness and consistency with historical observations. This approach more effectively captures short-range time dependencies, stabilizing the repair process and improving the accuracy of missing data repair, providing a basis for trend interpolation for continuous long missing data segments.

[0110] S4. Based on the third modality expansion matrix, a self-representation coefficient matrix and elastic network constraints are introduced to construct a periodic-scale periodic similarity regularization term.

[0111] Building energy consumption typically exhibits significant periodicity on a weekly scale. To further explore this periodic correlation, a periodic periodicity similarity regularization term is introduced. Although based on... Sparse representations of norms have been widely adopted, but one of their main limitations is the lack of grouping effects. Specifically, when there are strong pairwise correlations between weekly building energy consumption samples, L1 norm penalties often retain only one representative sample and suppress the remaining correlated samples, which may lead to poor repair performance. To overcome this problem, a self-representation method using elastic networks is employed to leverage the similarity between weekly building energy consumption data.

[0112] like Figure 5 As shown, this step is based on the third mode (cycle mode) expansion matrix, and introduces a self-representation coefficient matrix. To characterize the correlation between energy consumption data of different weeks; to add elastic network constraints (integrating L1 sparse constraints + L2 smooth constraints) to replace single sparse constraints; and to construct a periodic similarity regularization term at the week scale by combining the self-representation coefficient matrix and elastic network constraints.

[0113] The expression for the periodic-scale periodicity regularization term is:

[0114] ;

[0115] in, This is the third mode expansion matrix; For periodic similarity regularization at the weekly scale; This is a self-representation coefficient matrix used to characterize the similarity between different weeks. ; It is the Frobenius norm; It is an L1 norm; Here is the regularization strength parameter for elastic networks, used to represent the regularization strength of elastic networks. Elastic network penalty items; For elastic network mixing coefficients, used to control the tradeoff between sparsity and smoothness in the learned relation; The diagonal elements of the self-representation coefficient matrix are 0; Indicates constraints.

[0116] when As the regularization approaches 1, it becomes more sparsity-oriented; when... As the value approaches 0, it approaches the ridge regression (squared Frobenius) penalty, thus promoting the grouping effect among correlation coefficients. Compared with pure ridge regularization, the elastic network form can achieve a balance between sparsity and grouping effect, making peripheral similarity learning more stable and improving repair performance.

[0117] This embodiment constructs a multi-scale feature mining mechanism by forming a dual regularization of time autoregression and periodic similarity. Time autoregression compensates for the trend of long missing segments, while periodic similarity solves the information shortage caused by the lack of external covariates in single sensors. Elastic network constraints overcome the problem that single sparse constraints cannot handle strongly correlated samples, realize the synchronous extraction of features from adjacent weeks / similar weather weeks, and improve the accuracy of self-representation coefficients. The robustness of repairing high missing rates and complex missing patterns is greatly improved.

[0118] S5, integrate the low-rank constraint term, time autoregressive regularization term, and periodic-scale periodic similarity regularization term into a unified optimization objective function, and apply observation data consistency constraints.

[0119] This step integrates the S2 low-rank constraint term, the S3 time autoregressive regularization term, and the S4 periodic similarity regularization term into a unified optimization objective function. It combines the three types of constraints—low-rank, time-series, and periodic—into a unified optimization objective while ensuring the authenticity of the observed data. This achieves multi-constraint collaborative optimization, preserving the underlying low-rank structure of the data while conforming to the real time-series and periodic patterns of energy consumption. The repair logic is more in line with the characteristics of building energy consumption.

[0120] The expression for the unified optimization objective function is:

[0121] ;

[0122] in, This indicates finding the minimum value; Let be the three-dimensional low-rank tensor to be completed; A is the autoregressive coefficient matrix. It is a self-representation coefficient matrix; Assign a modal index number; For the first The weights corresponding to modal expansion; Representing a three-dimensional low-rank tensor Along the first The matrix of modal expansion; For the first The SCAD norm of the modal expansion matrix; The time-regression regularization strength parameter; For time-regressive regularization; The periodic similarity regularization intensity parameter is used for the periodic scale. For periodic similarity regularization at the weekly scale; Indicates constraints; For projection operators, preserve the tensor Set the observed element corresponding to the index to zero, and set all other elements to zero; It is the set of indices of the observed elements in the three-dimensional observation tensor; The three-dimensional observation tensor; This indicates that the diagonal elements of the self-representation coefficient matrix are 0.

[0123] Apply observation data consistency constraints to ensure that the repaired data matches the original data perfectly at known observation locations.

[0124] S6. The unified optimization objective function is solved iteratively using the alternating direction multiplier method, and auxiliary variables are introduced to correspond to each regularization term until the preset stopping condition is met, so as to obtain the complete building energy consumption data after the repair is completed.

[0125] Since directly optimizing the multidimensional variables in the unified optimization objective function is complex and often leads to unstable solutions and slow convergence, the problem can be decomposed into a series of simpler subproblems for iterative solving within the Alternating Direction Multiplier Method (ADMM) framework. In this framework, each iteration updates only one set of variables while fixing the rest, thus contributing to stable and efficient convergence. Given that each regularization term is obtained through... Tight coupling results in high-dimensionality and strong coupling of subproblems involved in each iteration. Therefore, variable splitting is performed by introducing auxiliary variables for each term.

[0126] Specifically, the alternating direction multiplier method is used to iteratively solve the unified optimization objective function. Auxiliary variables are introduced to correspond to each regularization term. The complex non-convex optimization is decoupled into multiple simple subproblems through variable splitting. Then, each subproblem is solved iteratively until the preset convergence stopping condition is met.

[0127] First, auxiliary tensors are introduced to correspond to each regularization term, transforming the multidimensional coupled optimization problem of the unified optimization objective function into an equivalent splitting form to reduce the coupling complexity between variables. The expression is as follows:

[0128] ;

[0129] in, These are the corresponding low-rank constraint terms with SCAD penalties. Matrix expanded along modes 1, 2, and 3 , , Auxiliary variables; The three-dimensional low-rank tensor to be completed; For the time autoregressive regularization term, the corresponding Auxiliary tensor; For the corresponding self-representation coefficient matrix Auxiliary variables used to handle constraints ; For the periodic similarity regularization term on a weekly scale, the corresponding Auxiliary variables; , , They are respectively , , The weights; In the periodic similarity regularization term at the week scale Auxiliary variables; It is the Frobenius norm; It is an L1 norm; Here is the regularization strength parameter for elastic networks, used to represent the regularization strength of elastic networks. Elastic network penalty items; The mixing coefficient of the elastic network; The diagonal elements of the self-representation coefficient matrix are 0; Indicates constraints; Representing auxiliary tensor Along the first The expansion matrix of the modalities.

[0130] Will , , After being treated as independent variable blocks, each mode can be updated separately under its own dimension and corresponding linear operator, thereby efficiently applying low-rank constraints during the ADMM iteration process.

[0131] Next, by introducing Lagrange multipliers and penalty parameters, the equality constraints are transformed into augmented Lagrange form, and the augmented Lagrange function is constructed, with the expression:

[0132] ；

[0133] in, To augment the Lagrange function; For Lagrange multipliers, ; For penalty parameters, , used to control the penalty strength of constraint terms in the augmented Lagrange function; Indicates the inner product;

[0134] Then, following a block descent strategy, during the iteration process, the update subproblems corresponding to each variable are solved sequentially to generate the update iteration values, expressed as:

[0135]

[0136] in, as auxiliary variables The The value is updated in the next iteration; Indicates the number of iterations; for Along the first The matrix of modal expansion Auxiliary variables; Represents the minimization operator; The corresponding variables are the first The value is updated in the next iteration; Indicates fixed division All variables outside of, for Perform a minimization solution; similarly , , , , They represent fixed divisions respectively. All variables outside of this.

[0137] When updating the auxiliary tensor, fix them separately. Under the condition of invariance, a closed update formula is obtained through singular value decomposition and SCAD shrinkage operator to efficiently realize low-rank constraints.

[0138] Among them, auxiliary tensor The update formula is:

[0139] ;

[0140] It can be equivalently rewritten using the completing the square method as follows:

[0141] ;

[0142] in, For nuclear norm; For penalty parameters; For the first In the next iteration, the corresponding Constrained Lagrange multipliers.

[0143] The update subproblem corresponds to the proximal mapping induced by the SCAD penalty term, and therefore a closed update can be obtained through SCAD-based singular value contraction.

[0144] make The singular value decomposition is represented as:

[0145] ;

[0146] in, , The first The left and right orthogonal matrices of the singular value decomposition in the next iteration; For the first One singular value; For matrix The smallest dimension; It is the transpose symbol;

[0147] The closed-form update formula for the auxiliary tensor obtained by applying the SCAD contraction operator to the singular values ​​is as follows:

[0148] ;

[0149] ;

[0150] in, For SCAD shrinkage operators; For modal folding operators;

[0151] When the iteration meets the preset stopping condition, the iteration stops and the repaired 3D building energy consumption tensor is output.

[0152] The preset stopping conditions include the residual being less than a preset threshold or reaching the maximum number of iterations.

[0153] Furthermore, the autoregressive coefficient matrix A is estimated by minimizing the sum of squared predicted residuals of all weekly sequences, and the coefficients of each weekly sequence are updated in parallel using the least squares method during the iterative process of the alternating direction multiplier method.

[0154] The Lagrange multipliers The update is based on the residual between the current repair tensor and the corresponding auxiliary variable, and is performed through gradient ascent.

[0155] The ADMM framework in this embodiment has good convergence and high computational efficiency. It transforms complex problems into easily solvable subproblems through variable splitting, reducing the difficulty of implementation. The decoupling of subproblems can support parallel computing and is suitable for the repair needs of large-scale building energy consumption monitoring systems.

[0156] In a preferred embodiment, the method of the present invention is verified using a publicly available "Hourly Energy Demand, Generation and Weather" data warehouse.

[0157] This dataset covers four years (January 2015 to December 2018) of hourly electricity demand, generation, market electricity prices, and meteorological data for Spain. It was initially compiled by ENTSO-E, a public portal for Transmission System Operator (TSO) data. The dataset exhibits significant temporal structure, making it suitable for time-series-based energy analysis. It contains a total of 35,064 hourly electricity demand records for the four years (2015: 8,760; 2016: 8,784; 2017: 8,760; 2018: 8,760).

[0158] In this embodiment, data from the first 16 weeks of 2016 were selected as the research sample, resulting in 2,688 hourly observations (16×7×24). Figure 6 As shown. To evaluate the robustness of the model to consecutive deletions, a random consecutive deletion pattern was constructed: the overall deletion rate was 10%, and the length of a single consecutive deletion segment was 25 hours.

[0159] To verify the superiority of the proposed univariate building energy consumption data restoration method, under the same missing data scenario, a low-rank tensor completion method based on the nuclear norm and the proposed SCAD-penalized low-rank tensor restoration method incorporating time-period regularization were used for restoration. The corresponding restoration results are as follows: Figure 7 As shown.

[0160] Both methods can repair missing building energy consumption data. However, compared to... Figure 7 Compared with the nuclear norm-based comparison method in (a), the proposed method performs better overall, especially in the area selected by the red box, where the improvement in repair effect is more significant. For quantitative comparison, the relative L2 error is used to measure the bias caused by missing entries, and the formula is as follows. The corresponding calculation results are summarized in Table 1.

[0161] ;

[0162] in, This is the relative L2 error; For the missing positions in the original data Actual energy consumption data; The set of indices for missing data, the set of observation locations The complement represents all the missing data locations that need to be repaired; For the model to the missing locations The repair results are the predicted energy consumption data.

[0163] Table 1. Relative L2 errors of the nuclear norm-based method and the proposed statistical method.

[0164]

[0165] As shown in Table 1, the relative L2 error of the nuclear norm-based method is 8.85%, while the method proposed in this invention reduces it to 4.90%, thereby significantly improving the accuracy of repairing missing building energy consumption data.

[0166] In summary, compared with the prior art, the present invention has the following beneficial effects:

[0167] (1) Single sensor adaptability: High-precision repair can be achieved solely through its own time series multi-scale modeling without the need for multiple sensors, weather, or other external covariates;

[0168] (2) Lower-rank constraints are more accurate: SCAD non-convex penalty replaces the traditional kernel norm, resulting in higher data fidelity;

[0169] (3) Stronger robustness to missing data: The dual constraints of short-term time series and long-term periodicity can cope with high missing data and long continuous missing data scenarios;

[0170] (4) Better engineering applicability: ADMM has high solution efficiency and can be parallelized, making it suitable for large-scale practical engineering applications.

[0171] Example 2

[0172] like Figure 8 As shown, the second embodiment of the present invention also provides a device for repairing low-rank single-sensor building energy consumption data, comprising:

[0173] The three-dimensional tensor reconstruction unit is used to collect one-dimensional time series of building energy consumption from a single sensor and perform three-dimensional tensor reconstruction according to hourly, daily and weekly scales to obtain three-dimensional observation tensors.

[0174] A low-rank constraint term construction unit is used to apply a smooth truncation absolute deviation penalty function to the matrix obtained by expanding the three-dimensional observation tensor along each mode to construct low-rank constraint terms;

[0175] The time autoregressive regularization term construction unit is used to construct an autoregressive model based on the third mode expansion matrix of the three-dimensional observation tensor along the circumferential mode expansion, and to construct a time autoregressive regularization term based on the autoregressive fitting residuals.

[0176] The weekly-scale periodic similarity regularization term construction unit is used to construct a weekly-scale periodic similarity regularization term based on the third mode expansion matrix, by introducing a self-representation coefficient matrix and elastic network constraints.

[0177] An optimization objective function unit is used to integrate the low-rank constraint term, the time autoregressive regularization term, and the periodic-scale periodic similarity regularization term into a unified optimization objective function, and to impose observation data consistency constraints.

[0178] The iterative solution unit is used to iteratively solve the unified optimization objective function using the alternating direction multiplier method, and introduces auxiliary variables corresponding to each regularization term until the preset stopping condition is met, so as to obtain the complete building energy consumption data after the repair is completed.

[0179] Example 3

[0180] The third embodiment of the present invention also provides a device for repairing low-rank single-sensor building energy consumption data, which includes a memory and a processor. The memory stores a computer program, which can be executed by the processor to realize the method for repairing low-rank single-sensor building energy consumption data as described above.

[0181] Example 4

[0182] The fourth embodiment of the present invention also provides a computer-readable storage medium storing computer-readable instructions, which, when executed by the processor of the device where the computer-readable storage medium is located, implement the method for repairing low-rank single-sensor building energy consumption data as described above.

[0183] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for repairing low-rank single-sensor building energy consumption data, characterized in that, include: S1 collects a one-dimensional time series of building energy consumption from a single sensor, and performs three-dimensional tensor reconstruction according to hourly, daily, and weekly scales to obtain the three-dimensional observation tensor; S2, Apply the smooth truncation absolute deviation penalty function to the matrix obtained by expanding the three-dimensional observation tensor along each mode to construct a low-rank constraint term; S3, construct an autoregressive model based on the third mode expansion matrix of the three-dimensional observation tensor along the circumferential mode expansion, and construct a time autoregressive regularization term based on the autoregressive fitting residual; S4. Based on the third modality expansion matrix, introduce the self-representation coefficient matrix and elastic network constraints to construct a periodic-scale periodic similarity regularization term; S5, integrate the low-rank constraint term, time autoregressive regularization term and weekly-scale periodic similarity regularization term into a unified optimization objective function, and apply observation data consistency constraints; S6, the unified optimization objective function is iteratively solved using the alternating direction multiplier method, and auxiliary variables are introduced to correspond to each regularization term until the preset stopping condition is met, so as to obtain the complete building energy consumption data after the repair is completed; Specifically, S2 is: First, the three-dimensional observation tensor is expanded into matrices along three modes, which correspond to the hourly, daily, and weekly scales, respectively. Singular value decomposition is performed on the expanded matrix of each mode to obtain the singular values ​​of each matrix; The Smooth Cut-Off Absolute Bias Penalty Function is applied to each singular value, and then combined with the weights of each mode and the projection constraints of the observed elements to construct a low-rank constraint term with SCAD penalty; the expression of the low-rank constraint term is: ； ； ； in, This indicates finding the minimum value; The three-dimensional low-rank tensor to be completed; The three-dimensional observation tensor; Assign a modal index number; express Along the first The matrix of modal expansion; For the first The weights corresponding to modal expansion; For the first The SCAD norm of the modal expansion matrix; Indicates constraints; For projection operators, preserve the tensor Set the observed element corresponding to the index to zero, and set all other elements to zero; It is the set of indices of the observed elements in the three-dimensional observation tensor; , These are the regularization parameters and adjustment parameters for SCAD penalty, respectively. for The One singular value; for The smallest dimension; For SCAD penalty function; These are the variables of the SCAD penalty function, i.e., singular values; The expression for the unified optimization objective function is: ； in, This is the autoregressive coefficient matrix; It is a self-representation coefficient matrix; The time-regression regularization strength parameter; For time-regressive regularization; The periodic similarity regularization intensity parameter is used for the periodic scale. For periodic similarity regularization at the weekly scale; This indicates that the diagonal elements of the self-representation coefficient matrix are 0.

2. The method for repairing low-rank single-sensor building energy consumption data according to claim 1, characterized in that... The expression for the time autoregressive regularization term is: ； in, This is the third mode expansion matrix; For time-regressive regularization; The order of the autoregressive model; , , These represent the hourly, daily, and weekly dimensions of the three-dimensional tensor, respectively. For the first The first line corresponding to the One autoregressive coefficient; The third mode matrix Line number Elements at each point in time.

3. The method for repairing low-rank single-sensor building energy consumption data according to claim 1, characterized in that... The expression for the periodic-scale periodicity regularization term is: ； in, This is the third mode expansion matrix; For periodic similarity regularization at the weekly scale; This is a self-representation coefficient matrix used to characterize the similarity between different weeks; It is the Frobenius norm; It is an L1 norm; Here is the regularization strength parameter for elastic networks, used to represent the regularization strength of elastic networks. Elastic network penalty items; The mixing coefficient of the elastic network; The diagonal elements of the self-representation coefficient matrix are 0; Indicates constraints.

4. The method for repairing low-rank single-sensor building energy consumption data according to claim 1, characterized in that... S6 specifically refers to: First, auxiliary tensors are introduced to correspond to each regularization term, transforming the multidimensional coupled optimization problem of the unified optimization objective function into an equivalent splitting form to reduce the coupling complexity between variables. The expression is as follows: ； in, These are the corresponding low-rank constraint terms with SCAD penalties. Matrix expanded along modes 1, 2, and 3 , , Auxiliary variables; The three-dimensional low-rank tensor to be completed; For the time autoregressive regularization term, the corresponding Auxiliary tensor; For the corresponding self-representation coefficient matrix Auxiliary variables used to handle constraints ; For the periodic similarity regularization term on a weekly scale, the corresponding Auxiliary variables; , , They are respectively , , The weights; In the periodic similarity regularization term at the week scale Auxiliary variables; It is the Frobenius norm; It is an L1 norm; Here is the regularization strength parameter for elastic networks, used to represent the regularization strength of elastic networks. Elastic network penalty items; The mixing coefficient of the elastic network; The diagonal elements of the self-representation coefficient matrix are 0; Indicates constraints; Representing auxiliary tensor Along the first The expansion matrix of the modes; , , Representing auxiliary variables Along the first The matrix of modal expansion, ; By introducing Lagrange multipliers and penalty parameters, the equality constraints are transformed into augmented Lagrange form, and the augmented Lagrange function is constructed, with the expression: ； in, To augment the Lagrange function; For Lagrange multipliers, ; For penalty parameters, , used to control the penalty strength of constraint terms in the augmented Lagrange function; Indicates the inner product; Next, following the block descent strategy, during the iteration process, the update subproblems corresponding to each variable are solved sequentially, as expressed in the following expression: ； in, as auxiliary variables The The value is updated in the next iteration; Indicates the number of iterations; for Along the first The matrix of modal expansion Auxiliary variables; Represents the minimization operator; The corresponding variables are the first The value is updated in the next iteration; Indicates fixed division All variables outside of, for Perform a minimization solution; When updating the auxiliary tensor, a closed update formula is obtained through singular value decomposition and SCAD shrinkage operator to efficiently realize low-rank constraints. Among them, auxiliary tensor The update formula is: ； It can be equivalently rewritten using the completing the square method as follows: ； in, For nuclear norm; For penalty parameters; For the first In the next iteration, the corresponding Constrained Lagrange multipliers; make The singular value decomposition is represented as: ； in, , The first The left and right orthogonal matrices of the singular value decomposition in the next iteration; For the first One singular value; For matrix The smallest dimension; It is the transpose symbol; The closed-form update formula for the auxiliary tensor obtained by applying the SCAD contraction operator to the singular values ​​is as follows: ； ； in, For SCAD shrinkage operators; For modal folding operators; When the iteration meets the preset stopping condition, the iteration stops and the repaired 3D building energy consumption tensor is output.

5. The method for repairing low-rank single-sensor building energy consumption data according to claim 4, characterized in that... The preset stopping conditions include the residual being less than a preset threshold or reaching the maximum number of iterations.

6. The method for repairing low-rank single-sensor building energy consumption data according to claim 1, characterized in that... The autoregressive coefficient matrix A is estimated by minimizing the sum of squared predicted residuals of all weekly sequences, and the coefficients of each weekly sequence are updated in parallel using the least squares method during the iterative process of the alternating direction multiplier method.

7. The method for repairing low-rank single-sensor building energy consumption data according to claim 4, characterized in that... The Lagrange multipliers The update is based on the residual between the current repair tensor and the corresponding auxiliary variable, and is performed through gradient ascent.

8. A device for repairing low-rank single-sensor building energy consumption data, used to implement the method for repairing low-rank single-sensor building energy consumption data as described in any one of claims 1-7, characterized in that, include: The three-dimensional tensor reconstruction unit is used to collect one-dimensional time series of building energy consumption from a single sensor and perform three-dimensional tensor reconstruction according to hourly, daily and weekly scales to obtain three-dimensional observation tensors. A low-rank constraint term construction unit is used to apply a smooth truncation absolute deviation penalty function to the matrix obtained by expanding the three-dimensional observation tensor along each mode to construct low-rank constraint terms; The time autoregressive regularization term construction unit is used to construct an autoregressive model based on the third mode expansion matrix of the three-dimensional observation tensor along the circumferential mode expansion, and to construct a time autoregressive regularization term based on the autoregressive fitting residuals. The weekly-scale periodic similarity regularization term construction unit is used to construct a weekly-scale periodic similarity regularization term based on the third mode expansion matrix, by introducing a self-representation coefficient matrix and elastic network constraints. An optimization objective function unit is used to integrate the low-rank constraint term, the time autoregressive regularization term, and the periodic-scale periodic similarity regularization term into a unified optimization objective function, and to impose observation data consistency constraints. The iterative solution unit is used to iteratively solve the unified optimization objective function using the alternating direction multiplier method, and introduces auxiliary variables corresponding to each regularization term until the preset stopping condition is met, so as to obtain the complete building energy consumption data after the repair is completed.

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