A fan fault diagnosis method based on small sample improved graph data structure

Abstract

The application discloses a fan fault diagnosis method based on small sample improved graph data structure, comprising the following steps: firstly, time domain vibration signals of fan rolling bearings are constructed into a proximity graph, then the proximity graph is optimized, an improved proximity graph is constructed, the improved proximity graph is optimized again, a feature index set representing different states of the fan rolling bearings is extracted, and finally, clustering analysis is performed on the feature index set, so that the fan rolling bearings in different states can be accurately classified. The improved proximity graph data structure pays more attention to local information of constructed data, and can more fully use data information of the fan rolling bearings; the improved proximity graph data structure is optimized by using sparse projection learning, so that redundant data and noise can be removed as much as possible, and the operation efficiency of the algorithm and the accuracy of feature extraction are improved; the feature index set representing different states of the fan rolling bearings can be effectively extracted, and the fan rolling bearings in different states can be accurately classified.

Description

Technical Field

[0001] This invention belongs to the field of wind turbine fault diagnosis technology, specifically relating to a wind turbine fault diagnosis method based on a small-sample improved graph data structure. Background Technology

[0002] Rolling bearings are a common and widely used critical component in wind turbines. Their failure directly impacts the operational status of machinery, ranging from minor shutdowns and production stoppages to serious accidents causing loss of life and property. During the cyclic rotation of a wind turbine's rolling bearing, localized damage to the inner ring, outer ring, or rolling elements generates periodic impact pulses at the points of contact with the damage. The bearing's fault signal is thus embedded in the vibration signal. Therefore, studying bearing vibration signals is of significant practical importance. During operation, bearings are affected by factors such as load and working environment, often resulting in their state signals being obscured by noise. Furthermore, the coupling of external environment, vibration source excitation, and response significantly increases the difficulty of feature extraction. Therefore, efficiently extracting fault impact signals is a key aspect of wind turbine rolling bearing fault diagnosis. Traditional fault feature extraction methods, including synchronous extraction transformation, empirical mode decomposition, local mean decomposition, and dual-channel signal-to-noise blind source separation, largely fail to achieve the goal of efficiently extracting fault feature indicators. As an emerging signal extraction method, graph signal processing (GSP) offers a novel approach to signal processing by studying data structures from a network perspective, gradually gaining attention from scholars both domestically and internationally. Derived from algebraic spectral graph theory, GSP aims to study the relationships between points within a graph, rather than simply analyzing image datasets and images themselves—a fundamental difference from image processing techniques. Currently, GSP is primarily used in image processing, chemistry, and machine learning. The key to GSP lies in the construction of the graph signal. Similarity matrices constructed from general graphs cannot fully utilize the local structural information contained in the samples, thus affecting subsequent feature selection operations. Furthermore, similarity matrices learned from raw data contain a large amount of irrelevant and redundant information, making it impossible to output a reliable feature selection model. Summary of the Invention

[0003] To address the technical problems existing in the prior art, the purpose of this invention is to provide a wind turbine fault diagnosis method based on a small-sample improved graph data structure.

[0004] To achieve the above objectives and technical effects, the technical solution adopted by this invention is as follows:

[0005] A wind turbine fault diagnosis method based on a small-sample improved graph data structure includes the following steps:

[0006] First, the time-domain vibration signal of the wind turbine rolling bearing is constructed into a proximity graph. Then, the proximity graph is optimized to construct an improved proximity graph. This improved proximity graph is then further optimized to extract a set of feature indicators representing different states of the wind turbine rolling bearing. Finally, cluster analysis is performed on the set of feature indicators to achieve accurate classification of wind turbine rolling bearings in different states.

[0007] Furthermore, the method includes the following steps:

[0008] 1) Obtain the time-domain vibration signal of the fan rolling bearing;

[0009] 2) Map the time-domain vibration signal obtained in step 1) to the graph domain, and construct a neighborhood graph using generalized graph learning;

[0010] 3) Optimize the neighborhood graph by using sparse feature projection learning to construct an improved neighborhood graph;

[0011] 4) Optimize the improved neighbor graph through sparse projection learning;

[0012] 5) Extract a set of characteristic indicators representing different states of the rolling bearings of the wind turbine;

[0013] 6) Cluster the set of feature indicators to achieve accurate classification of wind turbine rolling bearings in different states.

[0014] Furthermore, in step 1), the time-domain vibration signal of the wind turbine rolling bearing is acquired by an accelerometer.

[0015] Furthermore, in step 2), the formula for constructing the nearest neighbor graph is:

[0016]

[0017] In the formula, A ij Measuring the i-th sample x i and the j-th sample x j Similarity in the original feature space; the regularization parameter α can prevent the learned similarity matrix A from having trivial solutions and uniform distribution. If α→0, the sample will choose the nearest neighbor as its neighbor. If α→+∞, the similarity A between any samples is... ij = 1 / n; each row of the similarity matrix A i (i = 1, ..., n) are independent of each other, and the row vector A can be solved independently. i ;

[0018] The formula for constructing the nearest neighbor graph is further optimized as follows:

[0019]

[0020] In the formula, vector e i The j-th component If each sample has k nearest neighbor samples, the value of the regularization parameter α is automatically determined, and the calculation formula is as follows:

[0021]

[0022] In the formula, Represents vector e i The sequence obtained by arranging the elements in ascending order;

[0023] The formula for calculating the similarity matrix A is:

[0024]

[0025] Furthermore, in step 3), the feature projection learning formula is defined as follows:

[0026]

[0027] In the formula, W∈R d×c Let b represent the characteristic projection matrix, where b ∈ R. c Let F be the bias vector, and F = [F l F u ] T ∈R n×c This represents the predicted label matrix of the training samples, where γ is the regularization parameter.

[0028] Feature projection learning is introduced into the optimization of the neighborhood graph, and its formula is as follows:

[0029]

[0030] In the formula, the first term measures the nearest neighbor structure of the sample in the projected feature space;

[0031] If x i and x j Similarity S ij Since the value is relatively large, in the initial stage of algorithm optimization, let S = A to facilitate obtaining the true local structure of the data, and the regularization term... This avoids trivial solutions in S. The parameters θ and α are similar and are automatically determined based on the number of neighboring samples k.

[0032]

[0033] In the formula, For vectors The sequence of elements arranged in ascending order.

[0034] Furthermore, a smoothing constraint is applied to the predicted label F through label propagation, with the objective function being:

[0035]

[0036] In the formula, the first term represents the constraint of the sample nearest neighbor structure on the graph in the projected feature space; the second term avoids trivial solutions in S, and θ is the regularization parameter; the third term is a smoothing constraint defined on the predicted label F, so that nearest neighbor samples have similar predicted labels, L S Let λ be the Laplacian matrix defined on S, and λ be the regularization parameter; the fourth term enables the predicted label F of labeled samples to be... l Compared to its true label Y l To maintain consistency, B is a diagonal matrix. By setting its first l diagonal elements to a large constant, the guiding role of labeled samples in the algorithm can be fully utilized.

[0037] Furthermore, in step 4), an alternating iterative optimization method is used to solve for the objective function, which includes the target neighborhood map S, the predicted label F, the projection matrix W, the bias vector b, and the L of the projection matrix W. 2,1 Norm constraints.

[0038] Furthermore, the steps for solving the objective function include:

[0039] First, with the prediction label F and projection matrix W fixed, the objective function simplifies to:

[0040] min||X T W+1b T -F|| 2，1

[0041] Calculate the derivative of the above equation with respect to b, and set the derivative equal to 0 to directly obtain the optimal solution for b:

[0042] b T =(1 T F-1 T X T W) / n

[0043] Then, fixing the target neighborhood graph S and the projection matrix W, and substituting the optimal solution of b into the objective function, we obtain:

[0044] minλtr(F T L S F)+tr[(FY) T B(FY)]+β||HX T W-HF|| 2,1

[0045] In the formula, H = I-11 T / n, where I is the identity matrix; taking the derivative of F in the above equation and setting the derivative to 0, we get:

[0046] λLS F+B(FY)+2βZF-2βZX T W=0

[0047] Where Z = HD v H, D v Let be a diagonal matrix, and let i be the diagonal element.

[0048] Let P = (λL) S +B+2βZ) -1 Q = BY + 2βZX T W, calculate the optimal solution for predicting label F:

[0049] F = PQ

[0050] Fixing b and S, and substituting the optimal solution of F into the objective function, we get:

[0051]

[0052] The above equation can be simplified by the definition of the trace of a matrix as follows:

[0053] min 2tr(W T XL S X T W)+tr(2βW T XZX T W)-tr(Q T P T Q)+βγ||W|| 2,1

[0054] Let Q = BY + 2βZX T Substituting W into the above equation and calculating the derivative with respect to the projection matrix W, we obtain the optimal solution for the projection matrix W:

[0055] W=(E+γK) -1 C

[0056] In the formula, E = X(L) S / β+Z-2βZPZ)X T C = XZPBY, where K is the i-th diagonal element of the diagonal matrix K. ii =1 / 2(||w i ||+ε).

[0057] Finally, with the projection matrix W and the predicted label matrix F fixed, the objective function simplifies to:

[0058]

[0059] In this matrix S, each row is independent, therefore, the row vectors Si in S can be solved individually. i :

[0060]

[0061] Where, vector and The j-th component is Let scalar The above formula simplifies to:

[0062]

[0063] Furthermore, in step 5), the characteristic indicators of the wind turbine rolling bearing include the total variation of the graph signal, the energy index of the second graph, the maximum value of the characteristic value, and the graph structure connectivity.

[0064] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0065] (1) This invention proposes a wind turbine fault diagnosis method based on a small-sample improved graph data structure. Utilizing generalized graph learning and sparse projection learning, it effectively learns the local data structure of samples by fully leveraging the proximity information of samples in the projected feature space, thereby improving the efficiency and accuracy of fault diagnosis. Compared with existing time-domain feature extraction methods, this method can effectively extract a set of feature indicators characterizing different wind turbine rolling bearing states and accurately classify wind turbine rolling bearings in different states.

[0066] (2) The improved proximity graph data structure disclosed in this invention pays more attention to the local information of the data construction than ordinary graphs, and can make fuller use of the data information of the wind turbine rolling bearing.

[0067] (3) This invention utilizes sparse projection learning to optimize the improved neighbor graph data structure, which can make the improved neighbor graph data more effective, remove redundant data and noise as much as possible, and transform the learning of the improved neighbor graph data from the original data set to the projection space, thereby improving the computational efficiency of the algorithm and the accuracy of feature extraction. At the same time, it can realize the identification and prediction of the state of the wind turbine rolling bearing, thus providing theoretical and data support for the maintenance and repair strategy of the wind turbine rolling bearing. Detailed Implementation

[0068] The present invention will now be described in detail so that its advantages and features can be more easily understood by those skilled in the art, thereby providing a clearer and more explicit definition of the scope of protection of the present invention.

[0069] The following provides a brief overview of one or more aspects to offer a basic understanding of them. This overview is not an exhaustive summary of all conceived aspects, nor is it intended to identify key or decisive elements of all aspects, nor to define the scope of any or all aspects. Its sole purpose is to present some concepts of one or more aspects in a simplified form to prepare for the more detailed descriptions that follow.

[0070] A wind turbine fault diagnosis method based on a small-sample improved graph data structure can be summarized as follows:

[0071] First, the time-domain discrete signal of the wind turbine rolling bearing is constructed into a neighborhood graph signal using the generalized graph learning method. Then, the neighborhood graph signal is optimized using sparse feature projection learning. Next, the feature index set representing different states of the wind turbine rolling bearing is extracted by sparse projection optimization of the neighborhood graph signal. Finally, the K-median clustering algorithm is applied to evaluate and classify the feature index set to achieve the goal of accurate classification of wind turbine rolling bearings in different states.

[0072] The wind turbine fault diagnosis method based on a small-sample improved graph data structure disclosed in this invention specifically includes the following steps:

[0073] 1) The time-domain vibration signal of the rolling bearing of the wind turbine can be obtained by using an accelerometer, or the vibration signal of the rolling bearing of the wind turbine can be collected by other sensors.

[0074] 2) Map the time-domain vibration signal obtained in step 1) to the graph domain (essential), and construct a neighborhood graph using the generalized graph learning method.

[0075] Time-domain vibration signals obtained from accelerometers or other sensors are characterized by a small number of fault samples (i.e., small sample size) and limited sample label information. To fully utilize these characteristics, a neighborhood graph is constructed between samples to obtain their distribution information and highlight their local structural information. By training on the local distribution information of the samples, the data structure of the sample graph signal is improved, thereby enhancing the computational power of the feature extraction algorithm.

[0076] Considering the large amount of redundancy and noise in the original data feature space, using traditional neighborhood graph data structures can easily result in unreliable neighborhood graphs, affecting the ability of selected features to maintain the data distribution structure and the performance of feature selection. Furthermore, constructing traditional neighborhood graphs requires pre-setting the parameters of the similarity metric function, which limits the algorithm's practicality and versatility. To construct effective neighborhood graphs from small sample data, the neighborhood graph construction method is optimized as follows:

[0077]

[0078] In the formula, Aij The i-th sample x was measured i and the j-th sample x j In the original feature space, the regularization parameter α can prevent the learned similarity matrix A from having trivial solutions and a uniform distribution. Specifically, if α→0, a sample will choose its nearest neighbor as its neighbor; if α→+∞, the similarity A between any samples will be... ij = 1 / n. Because each row A in A... i (i = 1, ..., n) are independent of each other, and the row vector A can be solved independently. i Therefore, the nearest neighbor graph construction method can be optimized as follows:

[0079]

[0080] In the formula, vector e i The j-th component If each sample has k nearest neighbors, the value of the regularization parameter α can be automatically determined, and the calculation formula is as follows:

[0081]

[0082] In the formula, Represents vector e i The sequence is obtained by arranging the elements in ascending order. Therefore, the formula for calculating A is as follows:

[0083]

[0084] The aforementioned neighbor graph construction method can select k nearest neighbors for each sample in the original feature space and adaptively determine the similarity between the sample and its neighbors without using additional parameters. However, the original feature space contains a large amount of redundant and noisy information, and the neighbor information between samples is easily disturbed. Therefore, further optimization of the improved neighbor graph is needed.

[0085] 3) Optimize the neighbor graph signal using sparse feature projection learning (essential) to construct an improved neighbor graph.

[0086] Since feature projection can effectively mitigate the adverse effects of redundant and noisy features, the nearest neighbor relationships of samples in the projected feature space can more accurately describe the local distribution structure of samples. Therefore, feature projection is introduced into the optimization of the neighborhood graph.

[0087] Feature projection learning is defined by the following formula:

[0088]

[0089] In the formula, W∈R d×c Let b represent the characteristic projection matrix, where b ∈ R.c Let F be the bias vector, and F = [F l F u ] T ∈R n×c Let L represent the predicted label matrix of the training samples, and γ be the regularization parameter. The essence of feature projection learning is to transform L... 2,1 Applying sparsity constraints to the linear projection loss and the projection matrix W not only makes the model more robust but also enables the learned W to exhibit row sparse characteristics. Labeled (training set) and unlabeled (test set) samples are input into the linear regression model, and feature selection is achieved by simplifying the feature space by minimizing the loss between the feature projection subspace and the predicted label F.

[0090] The feature projection is incorporated into the optimization of the neighborhood graph, and its formula is as follows:

[0091]

[0092] In the formula, the first term measures the nearest neighbor structure of a sample in the projected feature space. Specifically, if x i and x j Similarity S ij It should be quite large.

[0093] In the initial stage of algorithm optimization, S can be set equal to A. This allows the graph model described above to more effectively acquire the true local structure of the data by learning the nearest neighbor relationships of samples in the projected feature space, aided by the nearest neighbor information between samples in the original feature space. Regularization term This can prevent trivial solutions from appearing in S, where the parameters θ and α are similar and can be automatically determined based on the number of neighboring samples k.

[0094]

[0095] In the formula, It is a vector The sequence of elements arranged in ascending order

[0096] To effectively combine the proposed improved nearest neighbor graph learning model and sparse projection learning, and to adaptively learn the local structure of samples in the projected feature space while selecting relevant feature subsets, a smooth constraint needs to be applied to the predicted label F through label propagation. The objective function is:

[0097]

[0098] In the formula, the first term represents the constraint of the sample nearest neighbor structure on the graph in the projected feature space; the second term avoids trivial solutions in S, and θ is the regularization parameter; the third term is a smoothing constraint defined on the predicted label F, so that nearest neighbor samples have similar predicted labels, L S Let λ be the Laplacian matrix defined on S, and λ be the regularization parameter; the fourth term enables the predicted label F of labeled samples to be... l Compared to its true label Y l To maintain consistency, B is a diagonal matrix. By setting its first l diagonal elements to a large constant, the guiding role of labeled samples in the algorithm can be fully utilized.

[0099] 4) Optimize the parameters of the improved proximity graph.

[0100] The objective function includes the target neighborhood map S, the predicted label F, the projection matrix W, and the bias vector b, as well as the L of the projection matrix W. 2,1 Norm constraints prevent direct optimization of the objective function and the solution parameters. Therefore, an alternating iterative optimization method is used to solve for the objective function.

[0101] First, with the prediction label F and projection matrix W fixed, the objective function can be simplified to:

[0102] min||X T W+1b T -F|| 2，1

[0103] Calculate the derivative of the above equation with respect to b, and set the derivative equal to 0 to directly obtain the optimal solution for b:

[0104] b T =(1 T F-1 T X T W) / n

[0105] Then, fixing the target neighborhood graph S and the projection matrix W, and substituting the optimal solution of b into the objective function, we obtain:

[0106] minλtr(F T L S F)+tr[(FY) T B(FY)]+β||HX T W-HF|| 2,1

[0107] In the formula, H = I-11 T / n, where I is the identity matrix. Taking the derivative of F in the above equation and setting the derivative to zero, we get: λL S F+B(FY)+2βZF-2βZX T W=0

[0108] Where Z = HD v H, D v Let be a diagonal matrix, and let i be the diagonal element.

[0109] Let P = (λL) S +B+2βZ) -1 Q = BY + 2βZX T W can be used to calculate the optimal solution for predicting the label F: F = PQ

[0110] Fixing b and S, and substituting the optimal solution of F into the objective function, we get:

[0111]

[0112] The above equation can be simplified to: by the definition of the trace of a matrix.

[0113] min2tr(W T XL S X T W)+tr(2βW T XZX T W)-tr(Q T P T Q)+βγ||W|| 2,1

[0114] Let Q = BY + 2βZX T Substituting W into the above equation and calculating the derivative with respect to W, we can obtain the optimal solution for W:

[0115] W=(E+γK) -1 C

[0116] In the formula, E = X(L) S / β+Z-2βZPZ)X T C = XZPBY, where K is the i-th diagonal element of the diagonal matrix K. ii =1 / 2(||w i ||+ε).

[0117] Finally, fixing W and F, the objective function simplifies to:

[0118]

[0119] In this matrix S, each row is independent, therefore, the row vectors Si in S can be solved individually. i :

[0120]

[0121] Where, vector and The j-th component is Let scalar The above formula can be simplified to:

[0122]

[0123] In summary, the iterative solution process for the objective function is as follows:

[0124] First, initialize S = A and F = Y; second, use fixed F and W to solve for b and F; then use the initialized diagonal matrix K to update the feature projection matrix W, and update K based on the current W, update the neighborhood graph S, and repeat the above optimization process until the objective function satisfies the convergence condition.

[0125] The above algorithm can learn the row vectors w of the feature projection matrix W through sparse projection. i (i = 1, ..., d) are associated with the corresponding features of the training sample X. Therefore, the i-th row vector of W directly measures the importance of the i-th feature in sample X (||w i The larger the value of ||2, the more important the corresponding feature. The row vector magnitude {||w} of the feature projection W is calculated. 1 ||2,...,||w d ||2}, and sort them from smallest to largest, and finally select the first r features as the output of feature selection.

[0126] 5) Optimize the Laplacian matrix of the neighboring graph through projection to extract graph indicators: total variation of the graph signal, SecondMohar indicator (second graph energy indicator), maximum value of eigenvalues ​​and Estrada indicator (graph structure connectivity) (other graph signal feature indicators can be extracted);

[0127] 6) Perform K-median clustering or other clustering methods on the dataset of fault characteristic indicators of wind turbine rolling bearings to achieve the purpose of feature extraction and classification of different faulty bearings.

[0128] Compared with the prior art, the present invention has at least the following advantages:

[0129] (1) Improved the problem of inaccurate extraction of traditional fault features (modal aliasing, endpoint effect);

[0130] (2) It solves the problem that the similarity matrix constructed from a general graph is difficult to reflect the local proximity information between samples;

[0131] (3) By using sparse projection learning to optimize and improve the neighbor graph data structure, the reliability of the graph signal data structure is improved, and feature indicators are accurately extracted and interference is eliminated.

[0132] Any parts or structures not specifically described in this invention can be made using existing technologies or products, and will not be elaborated upon here.

[0133] The above description is merely an embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural or procedural transformations made based on the content of the present invention specification, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of the present invention.

Claims

1. A wind turbine fault diagnosis method based on a small-sample improved graph data structure, characterized in that, Includes the following steps: 1) Obtain the time-domain vibration signal of the fan rolling bearing; 2) Map the time-domain vibration signal obtained in step 1) to the graph domain, and construct a neighborhood graph using generalized graph learning; 3) Optimize the neighborhood graph by using sparse feature projection learning to construct an improved neighborhood graph; 4) Optimize the improved neighbor graph through sparse projection learning; 5) Extract a set of characteristic indicators representing different states of the rolling bearings of the wind turbine; 6) Cluster the set of feature indicators to achieve accurate classification of wind turbine rolling bearings in different states; In step 2), the formula for constructing the nearest neighbor graph is: In the formula, Measurement of the first i Sample and the j Sample Similarity in the original feature space; regularization parameter Able to avoid learning similarity matrices A The existence of trivial solutions and uniform distributions, if The sample will select the nearest neighbor sample as its neighbor. Similarity between any samples Similarity matrix A Each line in They are independent of each other, and the row vectors can be solved individually. ; The formula for constructing the nearest neighbor graph is further optimized as follows: In the formula, vector The j Each component If each sample has k If the nearest neighbor samples are 1, the regularization parameter is automatically determined. The value of is calculated using the following formula: In the formula, Representing vectors The sequence obtained by arranging the elements in ascending order; Similarity matrix A The calculation formula is: 。 2. The wind turbine fault diagnosis method based on a small-sample improved graph data structure according to claim 1, characterized in that, In step 1), the time-domain vibration signal of the wind turbine rolling bearing is obtained by using an accelerometer.

3. The wind turbine fault diagnosis method based on a small-sample improved graph data structure according to claim 1, characterized in that, In step 3), the feature projection learning formula is defined as follows: In the formula, Represents the characteristic projection matrix, For bias vectors, This represents the predicted label matrix of the training samples. For regularization parameters; Feature projection learning is introduced into the optimization of the neighborhood graph, and its formula is as follows: In the formula, the first term measures the nearest neighbor structure of the sample in the projected feature space; if and similarity Larger, in the initial stage of algorithm optimization, let S = A This facilitates obtaining the true local structure of the data, and the regularization term... Able to avoid S A trivial solution exists, and the parameters... and Similarly, based on the number of neighboring samples k Automatic determination: In the formula, For vectors The sequence of elements arranged in ascending order. .

4. The wind turbine fault diagnosis method based on a small-sample improved graph data structure according to claim 3, characterized in that, Predicting labels through label propagation F Applying smoothing constraints, the objective function is: In the formula, the first term represents the constraint of the sample nearest neighbor structure in the projected feature space on the graph; the second term can avoid S A trivial solution appears. The first term is the regularization parameter; the second term is defined in the predicted label. F The smoothing constraint ensures that neighboring samples have similar predicted labels. Indicates that it is defined in S The Laplace matrix on, It is a regularization parameter; the fourth term enables the predicted label of labeled samples. Its real label Try to keep things consistent. B For a diagonal matrix, by changing its front... l The diagonal elements are set to a large constant to fully leverage the guiding role of labeled samples in the algorithm.

5. The wind turbine fault diagnosis method based on a small-sample improved graph data structure according to claim 4, characterized in that, An alternating iterative optimization method is used to solve the objective function, which includes a target neighborhood graph. S Predicted tags F Projection matrix W Sum of deviation vectors b and projection matrix W of Norm constraints.

6. The wind turbine fault diagnosis method based on a small-sample improved graph data structure according to claim 5, characterized in that, The steps for solving the objective function include: First, fix the predicted labels. F Projection matrix W The objective function simplifies to: Calculate the above formula about b The derivative of , and setting the derivative to 0, directly yields b The optimal solution: Then, fix the target proximity map. S and projection matrix W and will b Substituting the optimal solution into the objective function, we get: In the formula, , I It is the identity matrix; for the above formula F Taking the derivative and setting it to zero, we get: in, , Let be a diagonal matrix, and let its first... i diagonal elements ; make , Calculate the predicted label F The optimal solution: fixed b and S and will F Substituting the optimal solution into the objective function, we get: The above equation can be simplified by the definition of the trace of a matrix as follows: Will Substitute into the above equation and calculate the projection matrix. W The derivative of the derivative yields the projection matrix. W The optimal solution: In the formula, , diagonal matrix K The i diagonal elements . Finally, fix the projection matrix. W and predicted label matrix F The objective function simplifies to: Among them, matrix S Each row is independent of the others, therefore, they can be solved individually. S row vectors in : Where, vector and The j Each component is , Let scalar The above equation simplifies to: 。 7. The wind turbine fault diagnosis method based on a small-sample improved graph data structure according to claim 1, characterized in that, In step 5), the characteristic indicators of the wind turbine rolling bearing include the total variation of the graph signal, the energy index of the second graph, the maximum value of the characteristic value, and the graph structure connectivity.

Images