CN111144458A - Method for identifying mechanical faults under different working conditions of subspace embedded feature distribution alignment - Google Patents

Method for identifying mechanical faults under different working conditions of subspace embedded feature distribution alignment Download PDF

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CN111144458A
CN111144458A CN201911285714.0A CN201911285714A CN111144458A CN 111144458 A CN111144458 A CN 111144458A CN 201911285714 A CN201911285714 A CN 201911285714A CN 111144458 A CN111144458 A CN 111144458A
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陈仁祥
吴昊年
杨黎霞
李俊阳
张霞
唐林林
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Chongqing Jiaotong University
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Abstract

The invention relates to a method for identifying mechanical faults under different working conditions of subspace embedding feature distribution alignment. Firstly, aligning the source domain feature and the target domain feature in the target domain subspace by using a correlation alignment method to prevent domain deviation; then, a pseudo label is directly predicted for the target domain in the space training base classifier, and the pseudo label is used for quantitatively estimating respective weights of edge distribution and condition distribution of the two domains so as to adapt to the distribution difference of the source domain and the target domain; and finally, transmitting the learning rules of the two steps through a structural risk minimization frame, constructing a kernel function to establish a classifier, and iteratively updating to obtain a coefficient matrix of the final frame to finish fault diagnosis. The quantitative estimation of the respective weights of the two-domain edge distribution and the condition distribution has important significance in cross-domain mechanical fault diagnosis, and the feasibility and the effectiveness of the method are demonstrated through a multi-class composite fault diagnosis example. The invention is suitable for the fields of state monitoring, fault diagnosis and the like of mechanical equipment.

Description

Method for identifying mechanical faults under different working conditions of subspace embedded feature distribution alignment
Technical Field
The invention belongs to the technical field of mechanical detection, and relates to a method for identifying mechanical faults under different working conditions of subspace embedding feature distribution alignment;
background
As rotary machines become larger and more complex, the occurrence probability of compound failure increases. In practical application, the working condition of the rotating machinery changes greatly, and when a compound fault occurs, various fault characteristics are often interfered and coupled with each other, so that great challenge is brought to compound fault diagnosis. Especially, under different working conditions, the vibration signal has obvious non-stationarity due to the change of working condition parameters such as the rotating speed, the load and the like of the rotary machine, and the fault characteristics are difficult to effectively extract. In recent years, the migration learning is widely applied to the field of fault diagnosis under different working conditions. Luck and auspicious, etc. the fault of the gearbox under different working conditions is diagnosed by Transfer Component Analysis (TCA); the SSTCA method is applied to fault diagnosis of the rolling bearing under alternating working conditions. Longmingshan et al propose a Transfer Joint Matching (TJM) method, add feature selection on the basis of TCA, and adjust the internal distance and the class distance of a data class. The method adapts the two-domain edge probability distribution through cross-domain feature selection, and better results are obtained in the rotary machine diagnosis. And then, a Joint Distribution Adaptation (JDA) method is provided, and the edge probability Distribution of the heterogeneous Distribution data is equally adapted to the conditional probability Distribution, so that the data classification and identification tasks are better completed. When multi-class composite fault data under different working conditions are faced, the distribution difference causes that a training sample and test sample data are located in different subspaces, and the domain deviation cannot be effectively inhibited by simply adapting two-domain distribution in the nearest subspace. For Wang et al, a popular Embedded Distribution Alignment (MEDA) method is proposed, and an Embedded Manifold spatial feature Distribution Alignment method realizes degraded feature Alignment. However, it is far from sufficient to consider the feature selection problem, and the adaptation factor selection problem is faced when considering the data distribution. In the two-domain adaptation method involved in the transfer learning, the adaptation factor mu is extremely important, and the selection of the value directly determines whether the cross-domain features can be effectively transferred to obtain a better diagnosis result. In general, different values of μ can bring different cross-domain migration effects. For the adaptation factor, μ belongs to [0,1], μ → 0 (such as TCA/TJM) represents that the distribution distance of the source domain and the target domain is larger, and the adaptation of the edge probability distribution is more important; μ → 1 represents that the distance between the two domains is small and it is more important to adapt the conditional probability distribution. Especially when μ is 0.5 (as in JDA algorithms) both distributions are equally important. The existing method can only set a fixed value or manually adjust a mu value (such as a Balanced Distribution Addition (BDA) algorithm), and the selection of the values of different data needs to be manually selected after cross validation, so that the self-adaptive dynamic adjustment cannot be met.
Disclosure of Invention
In view of the above, the present invention provides a method for identifying mechanical faults under different working conditions of subspace embedding feature distribution Alignment, which aligns cross-domain features in a manner of aligning projection points in a target domain space by using a Correlation Alignment method (CORAL), and enhances adaptation performance by using an adaptation factor μ to adaptively adapt to edge probability and conditional probability distribution on the premise of effectively preventing domain deviation; and then transmitting The learning principle obtained in The two steps through a Structural Risk Minimization (SRM) framework, and constructing a final classifier for identification. For the quantitative distribution adaptation factor, the invention uses A-distance as a basic measurement value, and the adaptive quantitative estimation adaptation factor mu is used for dynamically adapting the importance of the two distributions, thereby achieving better effect.
In order to achieve the purpose, the invention provides the following technical scheme:
1) the spectrum of the original signal x (n) is obtained, the label corresponding to each category is used as input, and the solving formula of the spectrum signal is shown as the following formula:
Figure BDA0002317916840000021
where x (N) is a finite sequence of length N.
2) And calculating second-order statistics of the two-domain features, namely a covariance matrix to match distribution by using a correlation alignment method, aligning the feature distribution of the source domain and the target domain under the unsupervised condition, and effectively preventing domain deviation. It is represented by the following formula:
Figure BDA0002317916840000022
wherein ZSAnd ZTIs a source domain and target domain covariance matrix.
Figure BDA0002317916840000023
Is a covariance matrix of the transformed source domain features, A is a linear transformation matrix, | · |. the luminanceFIs Frobenious norm.
Further, the step 2) comprises the following steps
21) Will be provided with
Figure BDA0002317916840000024
Singular value transformation (SVD) is performed, and the specific formula is as follows:
Figure BDA0002317916840000025
u, U thereinTThe matrix is a left and right singular vector corresponding to the maximum singular value sigma S.
22) And then obtaining a transformation matrix A by utilizing a Roel-Penny pseudo-inverse solution, namely obtaining the optimal solution of the step 21), wherein the specific formula is as follows:
ATZSA＝UZT[1：r]∑S[1：r]UT ZT[1：r]
wherein r is the rank of the matrix where it is located, and [1: r ] is the maximum rank among the matrices 1 to r. The feature alignment transformation is completed.
3) And calculating to obtain a new sample of the source domain after the correlation alignment. Wherein the MMD distance of the maximum mean difference is distributed differently between the regions, and the MMD maximum mean difference value of the two domains is defined as:
Figure BDA0002317916840000031
Figure BDA0002317916840000032
representing a function transformed by features
Figure BDA0002317916840000033
Creating a regenerative nuclear hilbert space (RKHS), E [ ·]Representing the mean of the embedded samples. z is a radical ofsFor subspace feature aligned source domain new samples, ztIs a target domain sample.
4) And (3) calculating the distance between the two domains of MMD in the new transformation space to finally obtain a dynamic distribution alignment formula:
Figure BDA0002317916840000034
further, the step 4) includes the following steps
41) Obtaining an edge distribution alignment representation:
Figure BDA0002317916840000035
42) the conditional distribution alignment of each category is obtained as:
Figure BDA0002317916840000036
5) a quantitative distribution adaptation factor is calculated using a-distance as a base measure, which represents the error of the linear classifier used to distinguish the two domains. E (h) represents the error of the linear classifier h for distinguishing the source domain from the target domain. This gives:
dA(Ds，Dt)＝2(1-2∈(h))
thus, an edge distribution adapted A-distance value d is obtainedMDefining the adaptive A-distance value of each class condition distribution as dc，
Figure BDA0002317916840000037
Wherein
Figure BDA0002317916840000038
Representing a source domain sample with c classes.
6) Quantitative estimation adaptation factor estimation formula:
Figure BDA0002317916840000039
since the condition distribution estimate may be different each time, the dynamic distribution adaptive factor quantitative evaluation is updated after each iteration is completed.
7) Computing learning classifier f under minimum risk framework (SRM) using square loss l2As a loss factor evaluation criterion, the classifier f can be expressed as:
Figure BDA00023179168400000310
the classifier f is expressed using the central theorem as:
Figure BDA0002317916840000041
wherein β ═ (β)1，β2，...)T∈R(n+m)×1Represents a vector of coefficients of the image data,
Figure BDA0002317916840000042
represents a regenerated kernel hilbert space constructed from a kernel function K (·, ·), which is a kernel function. z (-) is a feature learning function obtained under the subspace correlation alignment method,
Figure BDA0002317916840000043
representing dynamic distribution alignment, Rf(-) as laplace regularization,
Figure BDA00023179168400000413
η, λ and ρ are the corresponding regularization parameters.
Further, the step 7) includes the following steps
71) In conjunction with step 7) the SRM framework is further expressed as:
Figure BDA0002317916840000044
wherein K ∈ R(n+m)×(n+m)Is a whole kernel matrix, and Kij＝K(zi，zj)，Aii∈R(n+m)×(n+m)Indicating the matrix for the diagonal domain when i ∈ DsWhen A isii1, otherwise Aii＝0。Y＝[y1，…，yn+m]For the target domain label matrix predicted from the source domain label, tr (-) represents the trace of the matrix.
72) Adding dynamic distribution adjustment, and further expressing the formula of the step 3) as follows according to the step 7) by using a central theorem and a kernel transformation skill:
Figure BDA0002317916840000045
wherein
Figure BDA0002317916840000046
The specific calculation matrix is as follows:
Figure BDA0002317916840000047
in the formula
Figure BDA0002317916840000048
And
Figure BDA0002317916840000049
73) a pairwise identity matrix is calculated. Using the similar geometric properties of the closest points in the subspace feature transform alignment, the pairwise identity-like matrix is represented as:
Figure BDA00023179168400000410
where sim (·,. cndot.) is a similarity function (e.g. cosine distance) used to measure the distance between two points, NP(zi) Is directed to ziP is the set of free parameters of this set in the method. Introducing a Laplace matrix L ═ D-W to obtain a diagonal matrix
Figure BDA00023179168400000411
74) The regularization formula is finally obtained as follows:
Figure BDA00023179168400000412
8) the classifier f expression is uniformly expressed as:
Figure BDA0002317916840000051
the partial derivatives of the signals are directly obtained,
Figure BDA0002317916840000052
the optimization can be obtained:
β＊＝((A+λM+ρL)K+ηI)-1AYT
thereby obtaining the entire classifier coefficient matrix.
The invention takes the frequency spectrum as input, utilizes a related alignment method to realize good domain deviation inhibition of feature alignment transformation in a target domain subspace, dynamically and adaptively adjusts the distribution weight of two domains, overcomes the interference of data distribution on a diagnosis result under different working conditions, establishes a classifier under a structure risk minimization frame, quantitatively evaluates an adaptation factor and avoids the diagnosis result fluctuation caused by manual setting. The method has the main advantages that: 1) feature alignment is completed in a target domain subspace, so that feature distortion caused by original space conversion is avoided, and domain deviation is effectively prevented; 2) the A-distance is used as a reference measurement value for quantitative estimation, except for the distribution adaptation factor mu, the iteration times of pseudo labels are reduced, and the influence of distribution difference on the composite fault identification is effectively reduced; 3) the classifier established under the SRM framework has strong classification and identification capabilities, and identification errors are reduced on the premise of stabilizing results.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.
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For the purposes of promoting a better understanding of the objects, aspects and advantages of the invention, reference will now be made to the following detailed description taken in conjunction with the accompanying drawings in which:
FIG. 1 is a flow chart of the algorithm of the present invention;
FIG. 2 is a spectrum diagram of various types of faults; (a) is the normal data frequency spectrum under 800 rpm; (b) the first type of fault frequency spectrum is at 800 rpm; (c) the frequency spectrum of the L4 fourth type fault under 800 rpm; (d) the frequency spectrum of the seventh fault at 800 rpm; (e) the frequency spectrum of the fourth type fault under 1500 rpm; (f) a seventh fault frequency spectrum at 1500 rpm; (g) the frequency spectrum of the eighth type fault under 1500 rpm; (h) the frequency spectrum of the ninth fault at 1500 rpm;
FIG. 3 is a graph showing a comparison of μ value selection;
FIG. 4 is a graph of cross-validation results for different methods; (a) comparing the accuracy of different methods during the migration from A to B; (b) comparing the accuracy of different methods during the migration from A to C; (c) comparing the accuracy of different methods when B is transferred to A; (d) comparing the accuracy of different methods during the migration from B to C; (e) comparing the accuracy of different methods during the migration from C to A; (f) the accuracy of different methods is compared when C is transferred to B.
Detailed Description
The embodiments of the present invention are described below with reference to specific embodiments, and other advantages and effects of the present invention will be easily understood by those skilled in the art from the disclosure of the present specification. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and examples may be combined with each other without conflict.
Wherein the showings are for the purpose of illustrating the invention only and not for the purpose of limiting the same, and in which there is shown by way of illustration only and not in the drawings in which there is no intention to limit the invention thereto; to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product; it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The same or similar reference numerals in the drawings of the embodiments of the present invention correspond to the same or similar components; in the description of the present invention, it should be understood that if there is an orientation or positional relationship indicated by terms such as "upper", "lower", "left", "right", "front", "rear", etc., based on the orientation or positional relationship shown in the drawings, it is only for convenience of description and simplification of description, but it is not an indication or suggestion that the referred device or element must have a specific orientation, be constructed in a specific orientation, and be operated, and therefore, the terms describing the positional relationship in the drawings are only used for illustrative purposes, and are not to be construed as limiting the present invention, and the specific meaning of the terms may be understood by those skilled in the art according to specific situations.
The following will describe in detail a preferred embodiment of the present invention, as shown in fig. 1.
1) Acquiring frequency spectrum information of the composite fault signal, wherein one working condition is used as a training sample, and the other working conditions are used as test samples and input into a transfer learning network;
2) initializing corresponding regularization parameters of a structure risk minimization frame η, such as 0.4, 0.01 and 0.4, setting a characteristic dimension d of the method to be 50, iteration times N to be 20 and proximity points p to be 30;
3) aligning two-domain second-order statistical features (covariance matrixes) in a target domain subspace through a CORAL method;
Figure BDA0002317916840000061
4) constructing a base classifier in the conversion subspace obtained in the step 3), and predicting a pseudo label for a target domain sample by using the marked data of the source domain
Figure BDA0002317916840000062
5) Constructing a kernel function K, where Z isS＝Z1：n，Zt＝Zn+1：m+1；
Figure BDA0002317916840000063
Wherein K ∈ R(n+m)×(n+m)As a whole kernel matrix
6) Repeating the steps 3), 4) and 5), and obtaining the adaptation factor of each iteration
Figure BDA0002317916840000071
And calculate M0And MC；
Figure BDA0002317916840000072
Figure BDA0002317916840000073
7) Continuously updating target domain pseudo-tags through iteration
Figure BDA0002317916840000074
And obtaining a final balance factor mu, obtaining a final classifier f, and completing the composite fault diagnosis.
Figure BDA0002317916840000075
The present example is illustrated below:
the first step is as follows: a fault signal to be analyzed is received. In the embodiment, an original signal x (n) is a vibration signal of a rolling bearing with a compound fault, the vibration signal of the compound fault of the bearing and a shaft is collected under different working conditions, and the operation rotating speed is 1) A working condition 800rpm respectively during collection; 2) b, working condition 1000 rpm; 3) c, working condition 1500 rpm; the loading load was 5 kg. The sampling frequency is 25600Hz, the sampling time is 10s, each fault signal is sampled for 4 times, and data samples are sequentially intercepted by taking 1024 as the number of analysis points. The pitch diameter of the bearing is 28.9052mm, 8 rolling elements, the contact angle is 9.08 degrees, and the diameter of the shaft is 19.05 mm. In the 10 fault types set in this example, 10 samples of a certain working condition are taken as training samples, and 10 samples of another working condition are taken as test samples. The spectrum is shown in FIG. 2, (a) is the normal data spectrum at 800 rpm; (b) the first type of fault frequency spectrum is at 800 rpm; (c) the frequency spectrum of the L4 fourth type fault under 800 rpm; (d) the frequency spectrum of the seventh fault at 800 rpm; (e) the frequency spectrum of the fourth type fault under 1500 rpm; (f) a seventh fault frequency spectrum at 1500 rpm; (g) the frequency spectrum of the eighth type fault under 1500 rpm; (h) the frequency spectrum of the ninth fault at 1500 rpm;
the second step is that: a, B, C three kinds of working condition data are cross-validated, and training set and test set are randomly extracted for 10 times. Wherein, 300 samples of each type of fault in the training set, 3000 samples of 10 types of faults in the training set, 200 samples of each type of fault in the testing set, and 2000 samples of 10 types of faults in the testing set. The network structure parameter is set as the characteristic dimension d is 50, the iteration number N is 10, p is 10, and the base classifier adopts KNN.
The third step: calculating second-order statistic of two-domain features, namely covariance matrix to match distribution, aligning feature distribution of source domain and target domain under unsupervised condition to obtain zsFor subspace feature aligned source domain new samples, ztIs a target domain sample;
the fourth step: calculating an adaptation factor for each iteration
Figure BDA0002317916840000076
And calculate M0And Mc；
The fifth step: iteratively updating target domain pseudo-label
Figure BDA0002317916840000077
Obtaining a final balance factor mu;
and a sixth step: and learning the characteristic alignment and distribution adaptation factor selection process by using a minimum risk frame, and constructing a final classifier to finish a classification recognition task.
Observing fig. 3, it can be known that the selection of μ directly affects the recognition accuracy, and in the conventional migration learning method, a fixed value is directly given to μ, or a value range is obtained according to a cross validation experiment. It is not possible to estimate an accurate value (e.g., TCA/JDA/BDA) from specific data. The SEDA allows quantitative estimation of the μ values with error accuracy as shown in table 1 and the detail diagram of fig. 3. In conclusion, the method provided by the invention has higher precision, faster convergence and better stability of accurate quantification of the mu value.
TABLE 1 μ value selection comparison results
Figure BDA0002317916840000081
Observing fig. 4, (a) is a comparison of the accuracy of different methods when a is migrated to B; (b) comparing the accuracy of different methods during the migration from A to C; (c) comparing the accuracy of different methods when B is transferred to A; (d) comparing the accuracy of different methods during the migration from B to C; (e) comparing the accuracy of different methods during the migration from C to A; (f) comparing the accuracy of different methods during the migration from C to B; the TCA method (mu is fixedly set to be 0.1) with single adaptive edge probability distribution has the worst identification effect, and the TJM method after the target domain selects the features has obviously better effect than the TCA. The BDA (artificially setting μ values according to data distribution types) considering the respective importance of the two distributions is superior overall to the JDA (μ value fixed set to 0.5) method in which the two distributions are treated equally and the TCA and TJM methods considering only edge distributions. Several methods with better identification precision are compared, and the classification precision is continuously improved along with the increase of the iteration times by combining the SEDA method shown in the figure 3. When N is 2, the accuracy of each group is more than 97%, the average accuracy is higher than that of the MEDA by 3 percentage points, the standard deviation is lower by 0.38 percentage point, the JDA and the BDA tend to be stable after 15 iterations, the SEDA identification accuracy is higher under the same iteration number, and the convergence is faster. This shows that the method provided by the invention not only has faster convergence, higher diagnosis efficiency, small deviation and good stability.
The results show that the average diagnosis rate of the SEDA reaches 97.80%, and better results are obtained within 5 iterations, which shows that the method is efficient and reliable for diagnosing the mechanical composite fault. Compared with JDA and TJM, which are limited by distribution adaptation, the result has larger fluctuation, BDA can adapt to the respective weights of two domains, but the distribution adaptation coefficient cannot be quantitatively estimated, so that the result is good and bad. In the data, due to the fact that the composite fault is formed by superposing the inner ring, the outer ring and the rolling body, if the characteristics of the two domains cannot be effectively aligned, the domain deviation is inhibited, the characteristic aliasing is caused, and the MEDA method is low in identification accuracy and only has 83.50% of the average diagnosis rate of weakness. The experiment further shows that for the compound fault diagnosis, the effective learning of the data characteristics is ensured, and the interference of the distribution under different working conditions on the data learning is overcome. The SEDA quantitatively estimates the distribution adaptive factors under the framework of minimizing the structural risk, adaptively adjusts the distribution of two domains, and has better stability, anti-interference capability and generalization performance.
Finally, the above embodiments are only intended to illustrate the technical solutions of the present invention and not to limit the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions, and all of them should be covered by the claims of the present invention.

Claims (4)

1. The method for identifying the mechanical faults under different working conditions of subspace embedding feature distribution alignment is characterized in that: the method comprises the following steps:
1) the spectrum of the original signal x (n) is obtained, the label corresponding to each category is used as input, and the solving formula of the spectrum signal is shown as the following formula:
Figure FDA0002317916830000011
wherein x (N) is a finite sequence of length N;
2) by utilizing a correlation alignment method, calculating second-order statistics of two-domain features, namely a covariance matrix to match distribution, aligning the feature distribution of a source domain and a target domain under the unsupervised condition, and effectively preventing domain deviation; represented by the formula:
Figure FDA0002317916830000012
wherein ZSAnd ZTA covariance matrix of a source domain and a target domain;
Figure FDA0002317916830000013
is a covariance matrix of the transformed source domain features, A is a linear transformation matrix, | · |. the luminanceFIs a Frobenious norm;
3) calculating to obtain a new sample of the source domain after the correlation alignment; wherein the MMD distance of the maximum mean difference is distributed differently between the regions, and the MMD maximum mean difference value of the two domains is defined as:
Figure FDA0002317916830000014
Figure FDA0002317916830000015
representing a function transformed by features
Figure FDA0002317916830000016
Creating a regenerative nuclear hilbert space (RKHS), E [ ·]Represents the mean of the embedded samples; z is a radical ofsFor subspace feature aligned source domain new samples, ztIs a target domain sample;
4) and (3) calculating the distance between the two domains of MMD in the new transformation space to finally obtain a dynamic distribution alignment formula:
Figure FDA0002317916830000017
5) calculating a quantitative distribution adaptation factor, using A-distance as a basic measurement value, the A-distance representing an error of a linear classifier for distinguishing two domains; e (h) represents the error of a linear classifier h for distinguishing a source domain from a target domain; obtaining:
dA(Ds，Dt)＝2(1-2∈(h))
obtaining the A-distance value d of edge distribution adaptationMDefining the condition distribution adaptation 4-distance value of each category as dc，
Figure FDA0002317916830000018
Wherein
Figure FDA0002317916830000019
Representing a source domain sample having c classes;
6) quantitative estimation adaptation factor estimation formula:
Figure FDA0002317916830000021
because the condition distribution estimated values may be different every time, the dynamic distribution adaptive factor quantitative evaluation is updated after each iteration is completed;
7) calculating learning classifier f under minimum risk frame SRM, and adopting square loss l2As a loss factor evaluation criterion, the classifier f is expressed as:
Figure FDA0002317916830000022
the classifier f is expressed using the central theorem as:
Figure FDA0002317916830000023
wherein β ═ (β)1，β2，...)T∈R(n+m)×1Represents a vector of coefficients of the image data,
Figure FDA0002317916830000024
represents a regenerated kernel hilbert space constructed by a kernel function K (·, ·), K being the kernel function; z (-) is a feature learning function obtained under the subspace correlation alignment method,
Figure FDA0002317916830000025
representing dynamic distribution alignment, Rf(-) as laplace regularization,
Figure FDA00023179168300000210
η, λ and ρ are the corresponding regularization parameters;
8) the classifier f expression is uniformly expressed as:
Figure FDA0002317916830000026
the partial derivatives of the signals are directly obtained,
Figure FDA0002317916830000027
the optimization can be obtained:
β*＝((A+λM+ρL)K+ηI)-1AYT
thereby obtaining the entire classifier coefficient matrix.
2. The method for identifying mechanical faults under different working conditions of subspace embedding feature distribution alignment according to claim 1, wherein the method comprises the following steps: the step 2) comprises the following steps:
21) will be provided with
Figure FDA0002317916830000028
Performing singular value transformation (SVD), wherein a specific formula is as follows:
Figure FDA0002317916830000029
u, U thereinTMaximum singular value of sigma of matrixSCorresponding left and right singular vectors;
22) and then obtaining a transformation matrix A by utilizing a Roel-Penny pseudo-inverse solution, namely obtaining the optimal solution of the step 21), wherein the specific formula is as follows:
ATZSA＝UZT[1：，r]∑S[1：r]UT ZT[1：r]
where r is the rank of the matrix in which it is located, [1: r ] is the maximum rank among the matrices 1 to r; completing the feature alignment transformation.
3. The method for identifying mechanical faults under different working conditions of subspace embedding feature distribution alignment according to claim 1, wherein the method comprises the following steps: the step 4) comprises the following steps:
41) obtaining an edge distribution alignment representation:
Figure FDA0002317916830000031
42) the conditional distribution alignment of each category is obtained as:
Figure FDA0002317916830000032
4. the method for identifying mechanical faults under different working conditions of subspace embedding feature distribution alignment according to claim 1, wherein the method comprises the following steps: the step 7) comprises the following steps
71) In conjunction with step 7) the SRM framework is further expressed as:
Figure FDA0002317916830000033
wherein K ∈ R(n+m)×(n+m)Is a whole kernel matrix, and Kij＝K(zi，zj)，Aii∈R(n+m)×(n+m)Indicating the matrix for the diagonal domain when i ∈ DsWhen A isii1, otherwise Aii＝0；Y＝[y1，…，yn+m]To mark according to the source domainLabeling a predicted target domain label matrix, tr (-) represents the trace of the matrix;
72) adding dynamic distribution adjustment, and further expressing the formula of the step 3) as follows according to the step 7) by using a central theorem and a kernel transformation skill:
Figure FDA0002317916830000034
wherein
Figure FDA0002317916830000035
The specific calculation matrix is as follows:
Figure FDA0002317916830000036
in the formula
Figure FDA0002317916830000037
And
Figure FDA0002317916830000038
73) calculating a pair identity matrix; using the similar geometric properties of the closest points in the subspace feature transform alignment, the pairwise identity-like matrix is represented as:
Figure FDA0002317916830000039
where sim (·,. cndot.) is a similarity function (e.g. cosine distance) used to measure the distance between two points, NP(zi) Is directed to ziP is the set of nearest points, P is the free parameter of this set in the method; introducing a Laplace matrix L ═ D-W to obtain a diagonal matrix
Figure FDA00023179168300000310
74) The regularization formula is finally obtained as follows:
Figure FDA0002317916830000041
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