Multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin

Abstract

This invention specifically relates to a multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margins, comprising: calculating the density scaling factor for each sample point using a density scaling factor algorithm; adaptively adjusting the neighborhood value of each sample point using the density scaling factor to obtain the ideal neighborhood value for each sample point; calculating its minimum linear representation weight to adaptively construct the intra-manifold graph and inter-manifold graph; calculating the corresponding intra-manifold graph scatter matrix and inter-manifold graph scatter matrix using the minimum linear representation error of the manifold graph; calculating the corresponding manifold margin using the intra-manifold graph scatter matrix and inter-manifold graph scatter matrix; and performing dimensionality reduction extraction on the original data by maximizing the manifold margin and minimizing intra-class differences to obtain the corresponding dimensionality-reduced data. This invention can adaptively adjust the neighborhood value of each sample point to the ideal neighborhood size through the density scaling factor, while better adapting to the characteristics of multi-manifold distributions in real-world data.

Description

Technical Field

[0001] This invention relates to the field of big data processing technology, specifically to a multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin. Background Technology

[0002] In recent years, advanced sensor technology, data acquisition equipment, and computer storage devices have enabled the accumulation of massive amounts of industrial big data. Effective processing and analysis of this multi-sensor big data allows for the early detection of potential faults and the identification of their location and severity, enabling the development of reasonable operation and maintenance plans to ensure the safe operation of rotating machinery. Large-scale industrial monitoring data contains rich information on equipment operation, helping to understand the operating conditions of internal components and thus determine the overall equipment status. However, large-scale data analysis presents new challenges for rotating machinery fault diagnosis research. The use of high-dimensional data for data mining and pattern recognition often leads to problems such as "data explosion" and the "curse of dimensionality." Accurately and efficiently extracting valuable information remains a key challenge in rotating machinery fault diagnosis technology research.

[0003] Feature reduction is an effective method for analyzing and processing the aforementioned problems. Its essence is to reduce the complexity and number of features in the original high-dimensional space by removing redundant or unimportant features or transforming and combining the original features into fewer features, thereby improving the performance of the data mining system. Since most data is nonlinear, linear dimensionality reduction algorithms such as principal component analysis, linear discriminant analysis, and multidimensional scaling are insufficient for handling high-dimensional nonlinear data. Therefore, research on dimensionality reduction using manifold learning has become a popular research topic. Manifold learning, with its ability to learn the nonlinear structure of the original data in high-dimensional space and discover essential dimensions, has gained attention from scholars both domestically and internationally in recent years.

[0004] Local Linear Embedding (LLE) is a classic algorithm for manifold learning. LLE uses the linear reconstruction relationship between a sample point and its local nearest neighbors as a characterization of the local geometric properties of the manifold. Its basic idea is to start with the local linearity of nonlinear data, transforming the global nonlinearity of the dataset into local linearity. Based on the global structural information provided by overlapping local neighborhoods, it seeks the globally optimal low-dimensional embedding representation while fully preserving the local geometric relationships of the data. LLE algorithms can learn local structures of arbitrary dimensions and have wide applications in many fields such as fault detection and image recognition. Numerous experts and scholars both domestically and internationally have conducted extensive research on the application and improvement of LLE algorithms.

[0005] In the process of using the LLE algorithm for dimensionality reduction, the neighborhood value is generally set to a fixed value for calculation. However, a fixed neighborhood size is only suitable for uniform manifold structures, while most real-world data are distributed on non-uniform manifolds. Therefore, a fixed neighborhood size is clearly unreasonable, leading to poor practicality of multi-manifold data dimensionality reduction. Furthermore, as a traditional manifold learning method, the LLE algorithm neglects modeling of data within and between manifolds, making it unable to handle the problem of multi-manifold identification of high-dimensional data, resulting in poor dimensionality reduction performance. Therefore, designing a method that can improve the practicality and effectiveness of multi-manifold data dimensionality reduction is an urgent technical problem to be solved. Summary of the Invention

[0006] To address the shortcomings of the existing technologies, the technical problem this invention aims to solve is: how to provide a multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin, which can adaptively adjust the neighborhood value of each sample point to the ideal neighborhood size through a density scaling factor, so as to better adapt to the uneven distribution of manifold structures in real data. At the same time, it can combine category labels and introduce the concept of manifold graph to model and describe data within and between manifolds, so as to better adapt to the characteristics of multi-manifold distribution of real data. Furthermore, by calculating the manifold divergence matrix and manifold margin, it can achieve feature dimensionality reduction extraction of high-dimensional feature datasets, thereby improving the practicality and effectiveness of multi-manifold data dimensionality reduction.

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

[0008] Multi-manifold feature reduction methods based on adaptive neighborhood selection and manifold margin include:

[0009] S1: For the original data containing several sample points, calculate the density scaling factor for each sample point using the density scaling factor algorithm;

[0010] S2: The ideal neighborhood value of each sample point is obtained by adaptively adjusting its neighborhood value through the density scaling factor of the sample point;

[0011] S3: Calculate the minimum linear representation error weight of each sample point using the ideal neighborhood value, so as to adaptively construct the corresponding manifold inner graph and manifold inter graph;

[0012] S4: Based on the manifold in-plot and manifold in-plot plots of each sample point, calculate the corresponding manifold in-plot scatter matrix and manifold in-plot scatter matrix using the minimum linear representation error of the manifold plot;

[0013] S5: Calculate the corresponding manifold margins using the scatter matrix of the manifold's internal graph and the scatter matrix of the manifold's external graph;

[0014] S6: Dimensionality reduction of the original data is performed by maximizing manifold margins and minimizing intra-class differences, and the corresponding dimensionality reduction data is used as the result of multi-manifold feature reduction.

[0015] Preferably, the density scaling factor of the sample points is calculated through the following steps:

[0016] S101: Calculate the local density of each sample point by the distance between sample points;

[0017] The formula is described as follows:

[0018] ;

[0019] like ,but ;

[0020] like ,but ;

[0021] In the formula: Represents sample points Local density; Represents sample points and The Euclidean distance between them; Indicates the set cutoff distance;

[0022] S102: Calculate the density scaling factor based on the local density of sample points;

[0023] The formula is described as follows:

[0024] ;

[0025] In the formula: This represents an exponential function with the natural constant e as its base. ;

[0026] ;

[0027] ;

[0028] In the formula: Represents sample points Density scaling factor; This represents the average density of the sample points; Represents the variance of the sample points; This indicates the number of sample points in the original data.

[0029] Preferably, the neighborhood values ​​of the sample points are adaptively adjusted through the following steps:

[0030] S201: Calculate its local density and neighborhood value of the sample points. value;

[0031] The formula is described as follows:

[0032] ;

[0033] In the formula: This represents the average density of the sample points; Represents sample points The neighborhood value; Represents sample points of value;

[0034] S202: Through Determine sample points Does the neighborhood value have extreme values? And make corresponding adjustments.

[0035] The formula is described as follows:

[0036] satisfy If the extreme value is 0, then there is no extreme value; otherwise, there is an extreme value.

[0037] This represents the average value of the density scaling factor;

[0038] When extreme values ​​exist:

[0039] like ,but ;

[0040] like ,but ;

[0041] S203: When there are no extreme values ​​in the neighborhood of a sample point, the neighborhood value is adaptively adjusted according to the density scaling factor of the sample point to increase the neighborhood value parameter of high-density sample points and decrease the neighborhood value parameter of low-density sample points.

[0042] The formula is described as follows:

[0043] ;

[0044] ;

[0045] In the formula: Indicates extreme value adjustment The result of this time; This represents the initial neighborhood value after adjusting for density and without extrema. The adjusted final ideal neighborhood value; This represents the constraint condition for determining whether the adaptively adjusted neighborhood value is the ideal neighborhood value; Represents sample points Density scaling factor.

[0046] Preferably, the minimum linear representation error weight of the manifold inner graph for each sample point is first calculated. Minimum linear representation of error weights between manifold graphs Then, based on the minimum linear representation error weight for each sample point, the corresponding minimum linear representation error weight matrix of the manifold inner graph is constructed. Minimum linear representation error weight matrix between manifold graphs Finally, the error weight matrix is ​​represented by the minimum linear representation of the manifold inner graph. Minimum linear representation error weight matrix between manifold graphs Calculate the scatter matrix of the graph within the manifold and the scatter matrix of the graph between manifolds;

[0047] Calculated using the following formula and :

[0048] ;

[0049] ;

[0050] In the formula: , Representing sample points respectively The minimum linear representation error weights of the manifold internal graph and the minimum linear representation error weights of the manifold external graph; , Representing sample points respectively The internal and external neighborhoods of the manifold; and The inverse matrix of the local Gram matrix; , , , The variable represented.

[0051] Preferably, the divergence matrix of the inner graph of the manifold and the divergence matrix of the inter-manifold graph are calculated using the following formulas:

[0052] ;

[0053] ;

[0054] ;

[0055] ;

[0056] In the formula: , Let represent the divergence matrix of the graph within the manifold and the divergence matrix of the graph between manifolds, respectively; , Let represent the minimum linear representation error weight matrix of the graph within a manifold and the minimum linear representation error weight matrix of the graph between manifolds, respectively. Represents the original data; Represents the identity matrix; This represents the transpose of the original data matrix; This indicates the matrix transpose.

[0057] Preferably, the manifold margin is calculated using the following formula:

[0058] ;

[0059] In the formula: Indicates the manifold margin; , Let represent the scatter matrix of the graph within the manifold and the scatter matrix of the graph between the manifolds, respectively.

[0060] Preferably, the dimensionality-reduced data is obtained through the following steps:

[0061] S601: Calculate the eigenvalues ​​and corresponding eigenvectors using the following eigenvalue decomposition formula, then sort the eigenvalues ​​from largest to smallest, and finally... The eigenvectors corresponding to the eigenvalues ​​are combined to form the transformation matrix. ;

[0062] The formula is described as follows:

[0063] ;

[0064] In the formula: In the eigenvalue decomposition problem A matrix consisting of the eigenvectors corresponding to the largest eigenvalues; This represents the eigenvalue to be determined; Represents the scatter matrix of the inner graph of the manifold; Indicates the manifold margin;

[0065] S602: By transforming the matrix Dimensionality reduction extraction is performed on the original data to obtain dimensionality-reduced data;

[0066] The formula is described as follows:

[0067] ;

[0068] In the formula: Represents dimensionality reduction data; Represents the original data; Represents the transformation matrix The transpose of .

[0069] Preferably, the eigenvalue decomposition formula is constructed through the following steps:

[0070] 1) Construct the following objective function to maximize manifold margins and minimize intra-class differences;

[0071] ;

[0072] In the formula: Representing dimensionality reduction data transpose;

[0073] 2) The objective function is transformed into the following single-objective optimization problem;

[0074] ;

[0075] In the formula: Represents the trace of a matrix;

[0076] 3) Linear transformation between original and embedded data And definition Therefore, the single-objective optimization problem is transformed into the following objective optimization constraint function;

[0077] ;

[0078] 4) Solve the objective optimization constraint function using the Lagrange multiplier method to obtain the following eigenvalue decomposition formula;

[0079] .

[0080] In the formula: In the eigenvalue decomposition problem A matrix consisting of the eigenvectors corresponding to the largest eigenvalues; This represents the eigenvalue to be determined; Represents the scatter matrix of the inner graph of the manifold; This represents the manifold margin.

[0081] The multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin in this invention has the following advantages compared with the prior art:

[0082] This invention calculates the density scaling factor for each sample point using a density scaling factor algorithm, and then adaptively adjusts the neighborhood value of each sample point to obtain the ideal neighborhood value. This allows for a full consideration of the relationship between the neighborhood value (the selection of nearest neighbors) and the density distribution of high-dimensional data. Under the assumption of satisfying the local linear hyperplane as much as possible, the neighborhood value of each sample point can be adaptively adjusted to the ideal neighborhood size through the density scaling factor. This reduces the sensitivity of existing manifold learning algorithms to neighborhood values, and thus better adapts to the uneven distribution of real-world data manifold structures, thereby helping to improve the practicality of dimensionality reduction for multi-manifold data.

[0083] This invention, based on determining the ideal neighborhood value of each sample point, adaptively constructs manifold in-manifold and inter-manifold graphs using the ideal neighborhood values ​​of sample points. It defines the manifold divergence matrix and manifold margin, and achieves dimensionality reduction extraction of the original data by maximizing the manifold margin and minimizing intra-class variance. This allows for modeling and describing in-manifold and inter-manifold data by incorporating the concept of manifold graphs in conjunction with class labels, better adapting to the characteristics of multi-manifold distributions in real-world data. Simultaneously, by calculating multi-manifold margins and manifold divergence matrices based on the modeling and description of in-manifold and inter-manifold data, the algorithm's ability to process multi-manifold distributed data is improved. Furthermore, the manifold margin and manifold in-manifold divergence matrix are optimized using Rayleigh entropy to find a subspace where data from different manifolds are more easily distinguishable, maximizing the manifold margin in the low-dimensional subspace and minimizing intra-class variance. This effectively achieves feature dimensionality reduction extraction from high-dimensional feature datasets, thereby improving the dimensionality reduction effect of multi-manifold data. Attached Figure Description

[0084] To make the objectives, technical solutions, and advantages of the invention clearer, the invention will now be described in further detail with reference to the accompanying drawings, wherein:

[0085] Figure 1 This is a schematic diagram of the local linear embedding algorithm;

[0086] Figure 2 This is a logic block diagram of the multi-manifold feature reduction method of the present invention;

[0087] Figure 3 A schematic diagram of the Swiss Roll dataset;

[0088] Figure 4 The following diagram illustrates the sensitivity of the LLE algorithm to the size of the neighborhood parameter: (a) represents k=6, (b) represents k=7, (c) represents k=8, and (d) represents k=16.

[0089] Figure 5 The effect of sample density on neighborhood size;

[0090] Figure 6 Flowchart for adaptive neighborhood selection;

[0091] Figure 7 This is a diagram of the gearbox vibration signal.

[0092] Figure 8 This is a dataset with a hybrid manifold structure.

[0093] Figure 9 A diagram showing the dimensionality reduction effect of multiple manifolds;

[0094] Figure 10 The dimensionality reduction results of four algorithms for the gearbox dataset are shown in the figure: △ represents normal data, ☆ represents pitting faults, ◇ represents wear faults, and ○ represents tooth breakage faults.

[0095] Figure 11 The recognition rate of the method of the present invention on a gearbox fault dataset with different initial neighborhood values ​​and dimensions;

[0096] Figure 12 The recognition rates of the three algorithms with different initial neighborhood values ​​on the gearbox fault dataset;

[0097] Figure 13 The highest recognition rate is given by different dimensionality reduction algorithms. Detailed Implementation

[0098] 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 some, not all, of the embodiments of the present invention. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. 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 only to illustrate 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.

[0099] It should be noted that similar reference numerals and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the figures, or the orientation or positional relationship commonly used when the product is in use. They are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first," "second," and "third," etc., are only used to distinguish descriptions and should not be construed as indicating or implying relative importance. In addition, the terms "horizontal," "vertical," etc., do not mean that the component is required to be absolutely horizontal or suspended, but can be slightly tilted. For example, "horizontal" only means that its direction is more horizontal than "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted. In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0100] The following detailed explanation illustrates the specific implementation methods:

[0101] Example:

[0102] Local Linear Embedding (LLE) is a classic algorithm for manifold learning. LLE uses the linear reconstruction relationship between a sample point and its local nearest neighbors as a characterization of the local geometric properties of the manifold. Its basic idea is to start with the local linearity of nonlinear data, transforming the global nonlinearity of the dataset into local linearity. Based on the global structural information provided by overlapping local neighborhoods, it seeks the globally optimal low-dimensional embedding representation while fully preserving the local geometric relationships of the data. LLE can learn local structures of arbitrary dimensions and has wide applications in many fields such as fault detection and image recognition. Numerous experts and scholars both domestically and internationally have conducted extensive research on the application and improvement of LLE. The LLE algorithm is illustrated in the diagram below. Figure 1 As shown, its algorithm mainly includes the following three steps:

[0103] 1) Finding nearest neighbors: The first step of LLE is based on the nearest neighbors in the input space. A certain metric between data points is selected, based on the distance from each sample point. The closest A point or a point within a fixed radius of a sample point is considered its nearest neighbor, forming a neighborhood set. .

[0104] 2) Calculate the optimal reconstruction weight matrix: For each sample point, perform linear reconstruction using its nearest neighbor data points. The reconstruction error of all data points can be expressed in the following mathematical form:

[0105] ;

[0106] In the formula Represents sample points and sample points The weights between them, if , .

[0107] 3) Obtaining low-dimensional embedding representation: The spatial positional relationship between all sample points and their nearest neighbors is preserved during the high-dimensional to low-dimensional mapping, that is, the linear reconstruction preserves... If the values ​​remain unchanged, the error in low-dimensional reconstruction is minimized.

[0108] ;

[0109] In the formula It is an identity matrix.

[0110] The optimal solution for this objective It can be achieved by calculating sparse, symmetric, positive semidefinite matrices. The smallest The eigenvectors corresponding to the non-zero eigenvalues ​​are obtained.

[0111] The LLE algorithm is a local processing method with advantages of simplicity, efficiency, and high runtime. However, in the process of dimensionality reduction using LLE, the neighborhood value is calculated using a fixed value. Real-world data is mostly distributed on non-uniform manifolds, so a fixed neighborhood size is clearly unreasonable. A fixed neighborhood size is only suitable for uniform manifold structures, resulting in poor practicality for multi-manifold data dimensionality reduction. Furthermore, as a traditional manifold learning method, LLE neglects modeling of data within and between manifolds, making it unable to handle the problem of multi-manifold identification of high-dimensional data, leading to poor dimensionality reduction results for multi-manifold data.

[0112] To address the above problems, the present invention discloses the following technical solution:

[0113] This embodiment discloses a multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin.

[0114] like Figure 2 As shown, a multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin includes:

[0115] S1: For the original data containing several sample points, calculate the density scaling factor for each sample point using the density scaling factor algorithm;

[0116] S2: The ideal neighborhood value of each sample point is obtained by adaptively adjusting its neighborhood value through the density scaling factor of the sample point;

[0117] S3: Calculate the minimum linear representation error weight of each sample point using the ideal neighborhood value, so as to adaptively construct the corresponding manifold inner graph and manifold inter graph;

[0118] S4: Based on the manifold in-plot and manifold in-plot plots of each sample point, calculate the corresponding manifold in-plot scatter matrix and manifold in-plot scatter matrix using the minimum linear representation error of the manifold plot;

[0119] S5: Calculate the corresponding manifold margins using the scatter matrix of the manifold's internal graph and the scatter matrix of the manifold's external graph;

[0120] S6: Dimensionality reduction of the original data is performed by maximizing manifold margins and minimizing intra-class differences, and the corresponding dimensionality reduction data is used as the result of multi-manifold feature reduction.

[0121] In this embodiment, fault data is divided into different fault categories according to fault category labels. After dimensionality reduction by the method of this invention, similar data become more compact and different categories of data become more dispersed, thus enabling fault classification to be achieved by combining with the SVM classification algorithm.

[0122] This invention calculates the density scaling factor for each sample point using a density scaling factor algorithm, and then adaptively adjusts the neighborhood value of each sample point to obtain the ideal neighborhood value. This allows for a full consideration of the relationship between the neighborhood value (the selection of nearest neighbors) and the density distribution of high-dimensional data. Under the assumption of satisfying the local linear hyperplane as much as possible, the neighborhood value of each sample point can be adaptively adjusted to the ideal neighborhood size through the density scaling factor. This reduces the sensitivity of existing manifold learning algorithms to neighborhood values, and thus better adapts to the uneven distribution of real-world data manifold structures, thereby helping to improve the practicality of dimensionality reduction for multi-manifold data.

[0123] This invention, based on determining the ideal neighborhood value of each sample point, adaptively constructs manifold in-manifold and inter-manifold graphs using the ideal neighborhood values ​​of sample points. It defines the manifold divergence matrix and manifold margin, and achieves dimensionality reduction extraction of the original data by maximizing the manifold margin and minimizing intra-class variance. This allows for modeling and describing in-manifold and inter-manifold data by incorporating the concept of manifold graphs in conjunction with class labels, better adapting to the characteristics of multi-manifold distributions in real-world data. Simultaneously, by calculating multi-manifold margins and manifold divergence matrices based on the modeling and description of in-manifold and inter-manifold data, the algorithm's ability to process multi-manifold distributed data is improved. Furthermore, the manifold margin and manifold in-manifold divergence matrix are optimized using Rayleigh entropy to find a subspace where data from different manifolds are more easily distinguishable, maximizing the manifold margin in the low-dimensional subspace and minimizing intra-class variance. This effectively achieves feature dimensionality reduction extraction from high-dimensional feature datasets, thereby improving the dimensionality reduction effect of multi-manifold data.

[0124] In practical implementation, the first step of the manifold learning algorithm is neighborhood selection, and appropriate neighborhood selection is a key factor for the algorithm's success. The method (algorithm) of this invention assumes that each sample point and its nearest neighbors lie in the same linear hyperplane. It uses the linear reconstruction relationship between the sample point and its local nearest neighbors as a characterization of the local geometric properties of the manifold. Its basic idea is to start with the local linearity of nonlinear data, transforming the global nonlinearity of the dataset into local linearity. Based on the global structural information provided by overlapping local neighborhoods, it seeks the globally optimal low-dimensional embedding representation while fully preserving the local geometric relationships of the data. Addressing the sensitivity issue of neighborhood selection in traditional manifold learning, and fully considering the relationship between the density distribution of high-dimensional data and the size of neighborhood values, a density scaling factor algorithm is proposed. This algorithm adaptively adjusts the neighborhood value of each sample point to the ideal neighborhood size using a density scaling factor, taking into full account any initial neighborhood value.

[0125] 1) Neighborhood selection sensitivity analysis

[0126] The choice of neighborhood size is crucial; it cannot be too small or too large. If the neighborhood is too small, the continuous topological space will be constructed into multiple separate subgraphs, causing some points to lose connection and failing to reflect global characteristics. If the neighborhood is too large, some sample data points will have neighborhoods from other folds. The relationship between the LLE dimensionality reduction effect and the neighborhood parameter size is as follows: Figure 3 and Figure 4 As shown, the dimensionality reduction effect of the LLE algorithm varies significantly with the different choices of the number of nearest neighbors. When the value of k is too small, although the linearity of the nearest neighbor region can be guaranteed, it cannot reflect the global properties of the data. When the value of k is too large, it destroys the premise of local linearity in manifold learning and uses linear processing methods for most nonlinearly distributed data, resulting in the destruction of the nonlinear structure between the original data.

[0127] 2) Sample density analysis

[0128] like Figure 5 As shown, considering the uneven distribution of manifold structure, in order to maximize the overlap between neighborhoods, facilitate information transmission, and protect the premise of local linearity, the neighborhood size should be larger in areas with high sampling density and smaller in areas with low sampling density.

[0129] Therefore, a density scaling factor algorithm is proposed based on the density distribution in high-dimensional space. The denser the distribution of sample points in high-dimensional space, the greater the density between samples, and the higher the density scaling factor. The larger the value, the looser the distribution of sample points in the high-dimensional space, resulting in a smaller density between samples; the larger the density scaling factor. The smaller.

[0130] Specifically, the density scaling factor for the sample points is calculated through the following steps:

[0131] S101: Calculate the local density of each sample point by the distance between sample points;

[0132] The formula is described as follows:

[0133] ;

[0134] like ,but ;

[0135] like ,but ;

[0136] In the formula: Represents sample points Local density; Represents sample points and The Euclidean distance between them; Indicates the set cutoff distance; sample points Local density This represents the number of data points whose distance is less than the cutoff distance.

[0137] The formula for calculating Euclidean distance is: ;

[0138] S102: Calculate the density scaling factor based on the local density of sample points;

[0139] The formula is described as follows:

[0140] ;

[0141] In the formula: This represents an exponential function with the natural constant e as its base. ;

[0142] ;

[0143] ;

[0144] In the formula: Represents sample points Density scaling factor; This represents the average density of the sample points; Represents the variance of the sample points; This indicates the number of sample points in the original data.

[0145] 3) Adaptive Neighborhood Selection

[0146] The density scaling factor for each sample point is calculated using the density scaling factor algorithm described above. The initial neighborhood values ​​are then adaptively adjusted based on the magnitude of the density scaling factor, increasing the neighborhood value parameter for high-density sample points and decreasing the neighborhood value parameter for low-density sample points. Considering that the extreme values ​​of the initial neighborhood values ​​have a significant impact on the adaptive result, the extreme values ​​of the initial neighborhood values ​​are initially adjusted before further adaptive adjustment to obtain the ideal neighborhood value.

[0147] Combination Figure 6 As shown, the neighborhood values ​​of sample points are adaptively adjusted through the following steps:

[0148] S201: Calculate its local density and neighborhood value of the sample points. value;

[0149] The formula is described as follows:

[0150] ;

[0151] In the formula: This represents the average density of the sample points; Represents sample points The neighborhood value; Represents sample points of value;

[0152] S202: Through Determine sample points Does the neighborhood value have extreme values? And make corresponding adjustments.

[0153] The formula is described as follows:

[0154] satisfy If the extreme value is 0, then there is no extreme value; otherwise, there is an extreme value.

[0155] When extreme values ​​exist:

[0156] like ,but ;

[0157] like ,but ;

[0158] This represents the average value of the density scaling factor;

[0159] S203: When there are no extreme values ​​in the neighborhood of a sample point, the neighborhood value is adaptively adjusted according to the density scaling factor of the sample point to increase the neighborhood value parameter of high-density sample points and decrease the neighborhood value parameter of low-density sample points.

[0160] The formula is described as follows:

[0161] ;

[0162] ;

[0163] In the formula: This indicates the extreme value adjustment in the previous step. The result of this time; This represents the initial neighborhood value after adjusting for density and without extrema. The adjusted final ideal neighborhood value; This represents the constraint condition for determining whether the adaptively adjusted neighborhood value is the ideal neighborhood value; Represents sample points Density scaling factor.

[0164] The method of this invention assumes that each sample point and its nearest neighbor are located in the same linear hyperplane. The linear reconstruction relationship between the sample point and its local nearest neighbor is used as a characterization of the local geometric properties of the manifold. The basic idea is to start from the local linearity of nonlinear data, transform the global nonlinearity of the dataset into local linearity, and provide global structural information based on the overlapping local neighborhoods. While fully preserving the local geometric relationship of the data, the method seeks the globally optimal low-dimensional embedding representation.

[0165] The present invention, through the above method, can fully consider the relationship between neighborhood value (selection of nearest neighbor points) and high-dimensional data density distribution. Under the assumption of satisfying the local linear hyperplane as much as possible, the neighborhood value of each sample point is adaptively adjusted to the ideal neighborhood size by the density scaling factor, which reduces the sensitivity of existing manifold learning algorithms to neighborhood value, and thus can better adapt to uneven multi-manifold data, thereby helping to improve the practicality of multi-manifold data dimensionality reduction.

[0166] In practical implementation, existing manifold learning methods neglect modeling of data within and between manifolds, making them unable to handle the problem of multi-manifold recognition of high-dimensional data. Considering that the minimum linear representation error in the LLE algorithm can reflect the degree of sample clustering to some extent, Professor Li Bo from Wuhan University of Science and Technology proposed the Locally Linear Representation of Manifold Margins (LLRMM) algorithm. This algorithm combines data labels to construct manifold-internal, inter-manifold, and complete manifold graphs, and uses this as a basis to measure multi-manifold margins. Furthermore, by maximizing manifold margins and preserving the local structure of multiple manifolds, it achieves feature dimensionality reduction extraction from high-dimensional feature datasets. This algorithm has been applied to feature extraction from face data with significant results. Considering that the goal of fault diagnosis is to achieve pattern recognition of fault types, reducing intra-class differences while increasing inter-class differences, this invention, using the inter-manifold spacing and manifold-internal graph scatter matrix defined by LLRMM, proposes an improved LLRMM algorithm more suitable for feature dimensionality reduction extraction from rotating machinery by maximizing manifold margins and minimizing intra-class differences, thus achieving feature dimensionality reduction extraction from high-dimensional feature datasets.

[0167] Specifically, the intra-manifold graph and inter-manifold graph are constructed by using the minimum linear representation error weights between sample point pairs.

[0168] In this embodiment, the manifold is constructed using minimum linear representation error weights. First, the minimum linear representation error weight for each sample point is calculated, and then combined with the sample category label to adaptively construct the manifold. Each row of the manifold (i.e., the minimum linear representation error weight matrix) is the minimum linear weight representation vector for a sample point. The minimum linear representation error of the sample points is used to represent the scatter matrix within the manifold and the scatter matrix between manifolds.

[0169] 1) Construction of manifold graph

[0170] The manifold inner graph reflects the local relationships within the manifold composed of similar data. For any sample point in the manifold inner graph... Select those that are of the same category and have The sample points with the smallest distance are used to form a sample point. The local neighborhood, therefore, within this neighborhood, similar to local linear embedding, sample points are used. The minimum linear error of its nearest neighbors of the same type is expressed as:

[0171] ;

[0172] In the formula: Sample points The inner nearest neighbor of the manifold.

[0173] For sample points The minimum linear representation of the error weights.

[0174] Because the minimum represents the condition that the sum of error weights is 1, the above formula can be rewritten as:

[0175] ;

[0176] By applying the Lagrange function, the local minimum linear representation error weights in the above objective function can be obtained, as shown in the following formula:

[0177] ;

[0178] In the formula: This represents sample points. The inner neighborhood of the manifold.

[0179] In the manifold graph, repeat the same steps for each point to obtain the minimum linear representation error weight matrix within the manifold. Similarly, by constructing the inter-manifold graph and the complete manifold graph, the local minimum linear representation error weights are shown in the formula above. The minimum linear representation error weight matrix of the inter-manifold graph can also be obtained. .

[0180] Specifically, the minimum linear representation error weight of the manifold inner graph for each sample point is first calculated. Minimum linear representation of error weights between manifold graphs Then, based on the minimum linear representation error weight for each sample point, the corresponding minimum linear representation error weight matrix of the manifold inner graph is constructed. Minimum linear representation error weight matrix between manifold graphs Finally, the error weight matrix is ​​represented by the minimum linear representation of the manifold inner graph. Minimum linear representation error weight matrix between manifold graphs Calculate the scatter matrix of the graph within the manifold and the scatter matrix of the graph between manifolds;

[0181] Calculated using the following formula and :

[0182] ;

[0183] ;

[0184] In the formula: , Representing sample points respectively The minimum linear representation error weights of the manifold internal graph and the minimum linear representation error weights of the manifold external graph; , Representing sample points respectively The internal and external neighborhoods of the manifold; and Denotes the inverse of the local Gram matrix. ; , , , The variable represents a value ranging from 1 to 1. .

[0185] In summary, the manifold inner graph and the manifold outer graph can be constructed using the minimum linear representation error weights between point pairs. In the manifold graph, any sample can be linearly represented by its nearest neighbors in the manifold inner graph, while the manifold divergence matrix represents the degree of clustering between data. Therefore, the minimum linear representation error of the manifold graph can be used to represent the manifold divergence, and thus the manifold inner graph divergence matrix and the manifold outer graph divergence matrix can be defined.

[0186] The divergence matrix of the inner graph and the divergence matrix of the inter-manifold graph are calculated using the following formulas:

[0187] ;

[0188] ;

[0189] ;

[0190] ;

[0191] in, and For the constructed matrix, such that , This represents the minimum linear representation error.

[0192] In the formula: , Let represent the divergence matrix of the graph within the manifold and the divergence matrix of the graph between manifolds, respectively; , Let represent the minimum linear representation error weight matrix of the graph within a manifold and the minimum linear representation error weight matrix of the graph between manifolds, respectively. Represents the original data; Represents the identity matrix; This represents the transpose of the original sample data matrix; This indicates the matrix transpose.

[0193] 2) Definition of manifold margins

[0194] To describe the degree of discretization between manifolds with different category labels, a new manifold margin is defined. Points on and other manifolds Distance minus manifold The interior distance of a manifold can be described as the degree of clustering or dispersion within the manifold, and can be represented by the manifold divergence matrix. Therefore, the manifold margin is defined as:

[0195]

[0196] In the formula: Indicates the first One manifold;

[0197] Indicates the first manifold The distance between it and other manifolds.

[0198] In the above formula, we define For the first manifold The sum of the minimum distances from every point on the manifold to all other manifolds, i.e.:

[0199] ;

[0200] In the formula: It is the first manifold Any sample point on the [database].

[0201] Sample points The minimum distance to other manifolds can be obtained from any sample point. It is represented by the square of the weighted average of its nearest neighbors in the manifold graph. Repeating this process for all sample points yields... The formula is expressed as follows:

[0202] ;

[0203] It can be seen that the above formula is actually the divination matrix between manifolds, so the manifold margins can be calculated using the following formula:

[0204] ;

[0205] In the formula: Indicates the manifold margin; , Let represent the scatter matrix of the graph within the manifold and the scatter matrix of the graph between the manifolds, respectively.

[0206] Based on the previously defined manifold intragraph scatter matrix, intergraph scatter matrix, and manifold margin, the purpose of the method in this invention is to find a subspace in which data from different manifolds are more easily distinguished, i.e., the manifold margin is maximized in the low-dimensional subspace and the intra-class difference is minimized.

[0207] The eigenvalue decomposition formula is constructed using the following steps:

[0208] 1) Based on the above analysis, to solve for the low-dimensional subspace, the two objective functions mentioned above need to be satisfied. That is, the following objective functions are constructed to maximize the manifold margin and minimize the intra-class variance;

[0209] ;

[0210] In the formula: Representing dimensionality reduction data transpose;

[0211] 2) To solve for the low-dimensional subspace, the two objective functions mentioned above need to be satisfied, which can be transformed into a single-objective optimization problem. That is, the objective function is transformed into the following single-objective optimization problem;

[0212] ;

[0213] In the formula: Represents the trace of a matrix;

[0214] 3) Generally speaking, traditional manifold learning methods encounter out-of-sample problems, thus requiring a linear relationship between the original data and the embedded data. And definition Therefore, the single-objective optimization problem is transformed into the following objective optimization constraint function;

[0215] ;

[0216] 4) The optimization objective is to find a low-dimensional subspace. The data after dimensionality reduction The goal is to maximize the manifold margins between different data categories and minimize the global manifold error. The Lagrange multiplier method is used to solve the constraint functions, yielding the eigenvalue decomposition formula and transformation matrix. From before eigenvalue decomposition The eigenvectors are formed by the eigenvalues ​​corresponding to the largest eigenvalues.

[0217] That is, by solving the objective optimization constraint function using the Lagrange multiplier method, the following eigenvalue decomposition formula is obtained;

[0218] .

[0219] In the formula: In the eigenvalue decomposition problem A matrix consisting of the eigenvectors corresponding to the largest eigenvalues; This represents the eigenvalue to be determined; Represents the scatter matrix of the inner graph of the manifold; This represents the manifold margin.

[0220] Specifically, the dimensionality-reduced data is extracted through the following steps:

[0221] S601: Calculate eigenvalues ​​and corresponding eigenvectors using the eigenvalue decomposition formula;

[0222] The formula is described as follows:

[0223] ;

[0224] This formula is the generalized eigenvalue decomposition formula in linear algebra (solving for generalized eigenvalues ​​in linear algebra). ), It is a generalized eigenvector.

[0225] In the formula: In the eigenvalue decomposition problem A matrix consisting of the eigenvectors corresponding to the largest eigenvalues; This represents the eigenvalue to be determined; Represents the scatter matrix of the inner graph of the manifold; Indicates the manifold margin;

[0226] S602: Sort the data feature values ​​from largest to smallest, and then... The transformation matrix is ​​formed by combining the eigenvectors corresponding to the data eigenvalues. ;

[0227] S603: By transforming the matrix Dimensionality reduction extraction is performed on the original data to obtain dimensionality-reduced data;

[0228] The formula is described as follows:

[0229] ;

[0230] In the formula: Represents dimensionality reduction data; Represents the original data; Represents the transformation matrix The transpose of .

[0231] Therefore, considering the contribution to the classification effect of the algorithm, we will look for a subspace in which data of different manifolds are more easily distinguished, maximize the distance between different manifolds, and minimize the distance between manifolds, that is, the manifold margin reaches its maximum in the low-dimensional subspace and the manifold internal graph scatter matrix reaches its minimum in the low-dimensional subspace.

[0232] Therefore, the improved algorithm steps of this invention are as follows:

[0233] Input: Raw data Data categories Initial number of nearest neighbors Dimensionality of low-dimensional space ;

[0234] Step 1: Using density adjustment factors Adaptively determine the neighborhood size for each data sample point ;

[0235] Step 2: Adaptively construct the inter-manifold graph and intra-manifold graph by adaptively selecting the neighborhood size;

[0236] Step 3: Calculate the weight matrices of the corresponding inter-manifold and intra-manifold graphs using the formula. And calculate the corresponding divergence matrix. ;

[0237] Step 4: Calculate the manifold margins based on the manifold divergence matrix. ;

[0238] Step 5: Solve the characteristic decomposition equation Sort the eigenvalues ​​from largest to smallest and then... The eigenvectors corresponding to the eigenvalues ​​are combined to form the transformation matrix. ,pass Solve to obtain the dimensionality-reduced data;

[0239] Output: Transformation matrix and the data after dimensionality reduction .

[0240] This invention, based on determining the ideal neighborhood value of each sample point, adaptively constructs the intra-manifold graph and inter-manifold graph using the ideal neighborhood value of the sample points. It defines the manifold divergence matrix and manifold margin, and achieves dimensionality reduction extraction of the original data by maximizing the manifold margin and minimizing intra-class differences. This enables the modeling of intra-manifold and inter-manifold data and uses this as a basis to measure the margins of multiple manifolds. At the same time, based on the definition of manifold margin and the calculation of the manifold divergence matrix, the optimization problem of the manifold learning algorithm is resolved, which can improve the recognition accuracy of the manifold learning algorithm. Furthermore, it can find a subspace through the manifold margin that makes the data of different manifolds easier to distinguish in this subspace, so that the manifold margin reaches its maximum in the low-dimensional subspace and the intra-class differences reach their minimum, thereby effectively realizing the feature dimensionality reduction extraction of high-dimensional feature datasets and improving the dimensionality reduction effect of multi-manifold data.

[0241] To better illustrate the advantages of the technical solution of the present invention, the following experiments are disclosed in this embodiment.

[0242] 1. Signal acquisition and preprocessing

[0243] This experiment uses a gearbox fault dataset. The gearbox data comes from a multi-stage centrifugal fan fault diagnosis unit, which consists of a variable-speed drive motor, bearings, gearbox, shaft, heavy-duty turntable, speed controller, etc., with a motor power of 0.75KW. Gearbox fault signals are acquired through a test bench, and the vibration signal sampling frequency is... Gear speed The module of the large gear Number of teeth pinion module Number of teeth The vibration signals include: normal signal of the large gear, pitting fault signal of the large gear, tooth breakage fault signal of the large gear, and wear fault signal of the small gear. The vibration signals and spectrum of the gearbox are as follows. Figure 7 As shown.

[0244] As can be seen from the time-domain and frequency spectrum diagrams, the gearbox vibration signal exhibits strong nonlinear and non-stationary characteristics, making it impossible to intuitively distinguish the differences between gear states. Therefore, to more comprehensively reflect the machine's health status, various signal processing methods are used to extract different feature parameters, such as time-domain features, frequency-domain features, and multifractal features. By dividing the gearbox vibration signal into several time data segments, the dynamic characteristics of each time segment are extracted to describe the signal's characteristics over the entire time period. Therefore, this experiment segmented the gearbox fault data, obtaining 120 samples, 4 categories, and 30 features. As shown in Table 1, this experiment selected 16 time-domain features, 5 frequency-domain features, and 9 multifractal features to construct a mixed-domain high-dimensional feature set.

[0245] Table 1 Feature Set of Mixed Domains

[0246]

[0247] 2. Analysis of the dimensionality reduction effect of multiple manifolds

[0248] To verify the dimensionality reduction effect of the method of the present invention on multiple manifolds, this experiment selects the SwissRoll dataset and the S-type dataset to construct a multi-manifold dataset for multi-manifold dimensionality reduction effect analysis, and selects KPCA, the classic manifold learning algorithm LLE and ALLRMM for comparison under the condition of ideal initial neighborhood values.

[0249] The experimental results are as follows:

[0250] like Figure 8 and Figure 9As shown, experimental results indicate that on multi-manifold datasets, the KPCA dimensionality reduction algorithm destroys the original topological structure after dimensionality reduction, failing to identify multiple manifolds. The traditional LLE (Large Manifold Learning) algorithm also fails to identify multiple manifolds after dimensionality reduction. However, the method of this invention (ALLRMM multi-manifold learning algorithm) preserves the structure of the multi-manifolds well. To further verify the contribution of different algorithms to classification, a gearbox fault dataset was selected for dimensionality reduction effect analysis. KPCA, the classic LLE algorithm, the LLRMM (Large Manifold Learning) algorithm, the improved LLRMM algorithm, and the ALLRMM algorithm were compared under ideal initial neighborhood value conditions.

[0251] The experimental results are as follows:

[0252] like Figure 10 As shown in (a), the original gearbox fault dataset is a result of selecting the most important 3D features based on the dimensionality reduction effect of a single feature. The fault modes exhibit severe aliasing, making direct identification impossible. After dimensionality reduction using KPCA, LLRMM, improved LLRMM, and ALLRMM (i.e., the method of this invention), the feature dimensionality reduction effect on the gearbox fault dataset is quite good, and the classification effect is significant. Compared to LLRMM and ALLRMM, KPCA dimensionality reduction results in some aliasing between different features, while LLE feature mapping shows significant aliasing between different fault mode features. Improved LLRMM, compared to ALLRMM, has comparable dimensionality reduction effects, both showing tighter data sharing among similar data, and less aliasing between different features compared to KPCA and LLE, resulting in better classification performance.

[0253] 3. Algorithm recognition accuracy analysis

[0254] To further compare the dimensionality reduction effects of these four algorithms, support vector machines were used to perform pattern recognition on the features extracted by these algorithms, and their recognition performance was demonstrated through experiments. In this experiment, two-thirds of the gearbox data samples were randomly selected to form the training set, and the remaining one-third formed the test set. To measure the recognition rate of the improved LLRMM algorithm under different initial neighborhood values ​​and dimensionalities, the initial neighborhood values ​​were adjusted. The values ​​are 2, 5, 10, 15, 20, and 25, representing the dimensionality reduction. Experiments were conducted for values ​​of 3, 5, 10, 15, and 20.

[0255] The experimental results are as follows:

[0256] like Figure 11 As shown, experimental results indicate that when the initial neighborhood value changes from 2 to 25, the recognition rate and the initial neighborhood value show a negative correlation and a decreasing trend. That is, the larger the initial neighborhood value, the lower the recognition rate, further demonstrating the sensitivity of the multi-manifold learning improved LLRMM algorithm to the selection of the initial neighborhood value. When the dimension... When changing from 2 to 20, the recognition rate and dimension... A positive correlation is shown. The graph shows that when… , When the improved LLRMM algorithm achieves a maximum average recognition rate of 97.50%, the ideal neighborhood value for the improved LLRMM algorithm is: .

[0257] To further compare the impact of different initial neighborhood values ​​on the algorithm, different initial neighborhood values ​​were selected for the LLE algorithm, the improved LLRMM algorithm, and the ALLRMM algorithm (i.e., the method of this invention) under the ideal feature dimension. To use the initial neighborhood value as a comparison, the average neighborhood value of all sample points after applying the adaptive neighborhood value adjustment algorithm under different initial neighborhood values ​​was calculated. The adjusted average neighborhood value was then compared and analyzed with the ideal neighborhood value. The experimental results are shown in Tables 2 and 3. Figure 12 , Figure 13 As shown.

[0258] Table 2 Different initial values Adaptive results of neighborhood values ​​under the given value

[0259]

[0260] Table 3 Average Recognition Accuracy of Gearbox Fault Data Set

[0261]

[0262] As shown in Table 2, the neighborhood values ​​adaptively adjusted by the ALLRMM algorithm proposed in this invention can be seen. When the initial neighborhood value is close to the ideal neighborhood value of 2, When the initial neighborhood values ​​are 20 or 25, they deviate significantly from the ideal neighborhood values. The ALLRMM algorithm effectively adjusts these values ​​to 3 or 2, bringing them closer to the ideal neighborhood values, demonstrating the stability and reliability of the adaptive neighborhood value adjustment algorithm. Figure 12 , 13 As shown in Table 3, the ALLRMM algorithm proposed in this invention achieved the highest average recognition rate for different initial neighborhood values. The LLE algorithm and the improved LLRMM algorithm achieved the highest average recognition rate for ideal neighborhood values. and The highest average recognition rates were achieved at 90.75% and 97.5%, respectively. However, from... Figure 12 It can be observed that the improved LLRMM and LLE algorithms have better initial neighborhood values. The values ​​are quite sensitive, and when the deviation from the ideal initial neighborhood value is large, the ALLRMM of the present invention shows a good recognition rate.

[0263] In summary, while mixed-domain features can effectively distinguish gearbox operating states, they still suffer from low recognition rates. The ALLRMM algorithm proposed in this invention adjusts the neighborhood value of each sample to near the ideal neighborhood value, reducing the sensitivity of traditional manifold learning algorithms to neighborhood values. Furthermore, referencing the manifold margin definition and manifold divergence matrix calculation of the LLRMM algorithm, the optimization problem is resolved, improving the algorithm's recognition accuracy. After feature extraction, the feature dimensionality is also reduced to some extent.

[0264] in conclusion

[0265] This invention first considers the relationship between the selection of nearest neighbors and sample density, and calculates the density scaling factor using the sample density of each data point to adaptively select the neighborhood size of the sample data. Then, it adaptively constructs the intra-manifold graph and inter-manifold graph through an adaptive neighborhood selection algorithm, and defines the manifold divergence matrix and manifold margin. Finally, it achieves feature extraction of the fault dataset by maximizing the manifold margin and minimizing the intra-class difference.

[0266] Experimental analysis results show that: 1) Single time-domain, frequency-domain, and multifractal domain features cannot completely distinguish the operating state of the gearbox. Mixed domain features can better distinguish the operating state of the gearbox, but the recognition rate is still low. 2) Traditional manifold learning methods only retain the local structural relationships between data and do not make a direct contribution to the classification of data. Therefore, the algorithm recognition efficiency is low, and traditional manifold learning cannot recognize multi-manifold distribution problems. 3) This invention improves the manifold graph construction through adaptive neighborhood selection, solving the sensitivity problem of neighborhood value selection in the LLRMM algorithm. For the rotating machinery fault dataset, combined with the manifold margin definition and manifold divergence matrix calculation of the LLRMM algorithm, an ALLRMM algorithm is proposed. Applying the ALLRMM algorithm to the gearbox fault dataset shows significant dimensionality reduction, good recognition accuracy, and makes the algorithm more universal and applicable in engineering.

[0267] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit the technical solutions. Those skilled in the art should understand that any modifications or equivalent substitutions to the technical solutions of the present invention without departing from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin, characterized in that, include: The different initial neighborhood values ​​of each sample point in the gearbox fault data are adjusted to ideal neighborhood values ​​to achieve dimensionality reduction of the gearbox fault data; Specifically: S1: For the original data containing several sample points, calculate the density scaling factor for each sample point using the density scaling factor algorithm; In step S1, the density scaling factor of the sample points is calculated through the following steps: S101: Calculate the local density of each sample point by the distance between sample points; The formula is described as follows: ； If then ; If then ; wherein: denotes the local density of sample points ; denotes the Euclidean distance between sample points and ; denotes a set cut-off distance; S102: Calculate the density scaling factor based on the local density of sample points; The formula is described as follows: ； ； ； where: denotes the density scaling factor for sample point ; denotes the average density of sample points; denotes the variance of sample points; denotes the number of sample points in the original data; S2: The ideal neighborhood value of each sample point is obtained by adaptively adjusting its neighborhood value through the density scaling factor of the sample point; In step S2, the neighborhood values ​​of the sample points are adaptively adjusted through the following steps: S201: Calculate the value of the sample point by the local density of the sample point and the neighborhood value value; The formula is described as follows: ； In the formulae: denotes the average density of the sample points; denotes the neighborhood value of the sample point ; denotes the value of the sample point ; ; S202: Through Determine sample points Does the neighborhood value have extreme values? And make corresponding adjustments. The formula is described as follows: satisfy If the extreme value is 0, then there is no extreme value; otherwise, there is an extreme value. This represents the average value of the density scaling factor; When extreme values ​​exist: like ,but ; like ,but ; S203: When there are no extreme values ​​in the neighborhood of a sample point, the neighborhood value is adaptively adjusted according to the density scaling factor of the sample point to increase the neighborhood value parameter of high-density sample points and decrease the neighborhood value parameter of low-density sample points. The formula is described as follows: ； ； In the formula: Indicates extreme value adjustment The result of this time; This represents the initial neighborhood value after adjusting for density and without extrema. The adjusted final ideal neighborhood value; This represents the constraint condition for determining whether the adaptively adjusted neighborhood value is the ideal neighborhood value; Represents sample points Density scaling factor; S3: Calculate the minimum linear representation error weight of each sample point using the ideal neighborhood value, so as to adaptively construct the corresponding manifold inner graph and manifold inter graph; S4: Based on the manifold in-plot and manifold in-plot plots of each sample point, calculate the corresponding manifold in-plot scatter matrix and manifold in-plot scatter matrix using the minimum linear representation error of the manifold plot; S5: Calculate the corresponding manifold margins using the scatter matrix of the manifold's internal graph and the scatter matrix of the manifold's external graph; S6: Dimensionality reduction of the original data is performed by maximizing manifold margins and minimizing intra-class differences, and the corresponding dimensionality reduction data is used as the result of multi-manifold feature reduction.

2. The multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin as described in claim 1, characterized in that, In step S4, the minimum linear representation error weight of the manifold inner graph for each sample point is first calculated. Minimum linear representation of error weights between manifold graphs Then, based on the minimum linear representation error weight for each sample point, the corresponding minimum linear representation error weight matrix of the manifold inner graph is constructed. Minimum linear representation error weight matrix between manifold graphs Finally, the error weight matrix is ​​represented by the minimum linear representation of the manifold inner graph. Minimum linear representation error weight matrix between manifold graphs Calculate the scatter matrix of the graph within the manifold and the scatter matrix of the graph between manifolds; Calculated using the following formula and : ； ； In the formula: , Representing sample points respectively The minimum linear representation error weights of the manifold internal graph and the minimum linear representation error weights of the manifold external graph; , Representing sample points respectively The internal and external neighborhoods of the manifold; and The inverse matrix of the local Gram matrix; , , , The variable represented.

3. The multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin as described in claim 2, characterized in that, The divergence matrix of the inner graph and the divergence matrix of the inter-manifold graph are calculated using the following formulas: ； ； ； ； In the formula: , Let represent the divergence matrix of the graph within the manifold and the divergence matrix of the graph between manifolds, respectively; , Let represent the minimum linear representation error weight matrix of the graph within a manifold and the minimum linear representation error weight matrix of the graph between manifolds, respectively. Represents the original data; Represents the identity matrix; This represents the transpose of the original data matrix; This indicates the matrix transpose.

4. The multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin as described in claim 3, characterized in that, In step S5, the manifold margins are calculated using the following formula: ； In the formula: Indicates the manifold margin; , Let represent the scatter matrix of the graph within the manifold and the scatter matrix of the graph between the manifolds, respectively.

5. The multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin as described in claim 4, characterized in that, In step S6, the dimensionality-reduced data is extracted through the following steps: S601: Calculate the eigenvalues ​​and corresponding eigenvectors using the following eigenvalue decomposition formula, then sort the eigenvalues ​​from largest to smallest, and finally... The eigenvectors corresponding to the eigenvalues ​​are combined to form the transformation matrix. ; The formula is described as follows: ； In the formula: In the eigenvalue decomposition problem A matrix consisting of the eigenvectors corresponding to the largest eigenvalues; This represents the eigenvalue to be determined; Represents the scatter matrix of the inner graph of the manifold; Indicates the manifold margin; S602: By transforming the matrix Dimensionality reduction extraction is performed on the original data to obtain dimensionality-reduced data; The formula is described as follows: ； In the formula: Represents dimensionality reduction data; Represents the original data; Represents the transformation matrix The transpose of .

6. The multi-manifold feature reduction method based on adaptive neighborhood selection and manifold margin as described in claim 5, characterized in that, The eigenvalue decomposition formula is constructed using the following steps: 1) Construct the following objective function to maximize manifold margins and minimize intra-class differences; ； In the formula: Representing dimensionality reduction data transpose; 2) The objective function is transformed into the following single-objective optimization problem; ； In the formula: Represents the trace of a matrix; 3) Linear transformation between original and embedded data And definition Therefore, the single-objective optimization problem is transformed into the following objective optimization constraint function; ； 4) Solve the objective optimization constraint function using the Lagrange multiplier method to obtain the following eigenvalue decomposition formula; ； In the formula: In the eigenvalue decomposition problem A matrix consisting of the eigenvectors corresponding to the largest eigenvalues; This represents the eigenvalue to be determined; Represents the scatter matrix of the inner graph of the manifold; This represents the manifold margin.

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