Abstract
The high dimensionality data originating from rolling bearing measuring signals with nonlinearity and low signal to noise ratio often contains too much disturbance like interference and redundancy for accurate condition identification. A novel manifold learning named Local coordinate weight reconstruction (LCWR) is proposed to remove such disturbance. Due to the different contribution of samples to their manifold structure, weight value is used for the contribution difference. By reconstructing local lowdimensional coordinates according to its weight function about geodesic distance in neighborhood, LCWR targets to reduce reconstruction error, preserve intrinsic structure of the high dimensionality data, eradicate disturbance and extract sensitive features as global lowdimensional coordinates. The experimental results show that the intraclass aggregation and interclass differences of global lowdimensional coordinates extracted via LCWR are better than those of local tangent space alignment (LTSA), locally linear embedding (LLE) and principal component analysis (PCA). The accuracy reaches the highest 96.43 % using SVM to identify LCWR based global lowdimensional coordinates, and its effectiveness is testified in diagnosis of rolling bearing fault.
Highlights
 A manifold learning named Local coordinate weight reconstruction (LCWR) is proposed for nonlinear dimensionality reduction.
 LCWR targets to eradicate disturbance and extract sensitive features from the high dimensionality data.
 The intraclass aggregation and interclass differences of global lowdimensional coordinates via LCWR are better than other methods.
 The accuracy reaches the highest 96.43% and testifies the effectiveness of LCWR in diagnosis of rolling bearing fault.
1. Introduction
Rolling bearing plays a key role in rotating machinery. It is necessary to monitor rolling bearing condition and identify its fault to avoid accident [13]. The state information of rolling bearing is usually described with high dimensionality data consisting of multiple characteristics in time and frequency domain [46], which contains redundancy and interference, and exists nonlinearity. Therefore, several works have been explored to remove such disturbance and obtain lowdimensional sensitivity features for better accuracy and efficiency of rolling bearing fault [7, 8].
As a type of classic manifold learning for dimensionality reduction [9, 10], local tangent space alignment (LTSA) completing nonlinear dimensionality reduction through finding out neighborhoods of high dimensionality samples, carrying out local dimensionality reduction, and realigning all neighborhoods’ lowdimensional coordinates to construct global lowdimensional coordinates [11], recently has been used for fault diagnosis besides its earlier successful applications to image processing, data mining, machine learning, etc. [12, 13]. For instance, Zhang et al. proposed supervised locally tangent space alignment (SLTSA) to optimize the neighborhood selection of LTSA based on the training samples’ categories, so that the neighborhood includes the same samples as possible to accurately reflect the local structures of different types of bearing fault signals [14]. Li et al. improved the accuracy of bearing fault diagnosis using LLTSA dimensionality reduction [15]. Kumar A and Kumar R utilized Linear Local Tangent Space Alignment (LLTSA) to suppress noise and retain characteristic defect frequencies of rolling bearing with inner and ball fault [16]. Su et al. proposed orthogonal supervised linear local tangent space alignment (OSLLTSA) to make the neighborhood selection of LLTSA better by introducing sample’s label information, which removed interference and redundancy in high dimensionality fault data and extracted lowdimensional sensitivity fault features [17]. Wang et al. proposed supervised incremental local tangent space alignment (SILTSA) through embedding supervised learning into the incremental local tangent space alignment to extract bearing fault characteristics, process new samples and classify [18]. Su et al. stated supervised extended local tangent space alignment (SELTSA) to enhance intraclass aggregation and interclass differences of nonlinear high dimensionality samples by defining distance between samples and optimizing neighborhood choice based on class label [19]. In summary, the current interests focused on neighborhood optimization options and local tangent space estimation of LTSA for dimensionality reduction and lowdimensional sensitivity fault characteristics extraction.
Different from above methods, local coordinate weight reconstruction (LCWR) manifold learning is proposed to reconstruct local coordinates by weight coefficient, so as to extract global lowdimensional sensitivity features for improving fault diagnosis capability of rolling bearing. The following is as below: Section 2 proposes LCWR for coordinate reconstruction, Section 3 validates LCWR and Section 4 is conclusions.
2. LCWR
LCWR has two major tasks. First, the projection coordinates of k nearest neighbors of each sample on the tangent space of the neighborhood are calculated to build the local lowdimensional manifold using LTSA. Next, the global lowdimensional coordinates are obtained by aligning the lowdimensional coordinates of all neighborhoods according to weight coefficient, which is a main innovation of LCWR.
2.1. Local coordinate computation
Let sample matrix $\mathbf{X}=\left({\mathbf{x}}_{1},{\mathbf{x}}_{2},\dots ,{\mathbf{x}}_{N}\right)\in {\mathbf{R}}^{D\times N}$, $D$ is the number of sample dimensions and $N$ is the number of samples. Sample ${\mathbf{x}}_{i}$ and its $k$ nearest samples (including ${\mathbf{x}}_{i}$) constitute local neighborhood matrix ${\mathbf{X}}_{i}=({\mathbf{x}}_{i1},{\mathbf{x}}_{i2},\dots ,{\mathbf{x}}_{ik})\in {\mathbf{R}}^{D\times k}$, each neighborhood exists a local tangent space ${\mathbf{Q}}_{i}\in {\mathbf{R}}^{D\times d}$$(d<D)$ consisting of standard orthogonal basis vectors, and ${\mathbf{\Theta}}_{i}=\left({\mathbf{\theta}}_{i1},{\mathbf{\theta}}_{i2},\dots ,{\mathbf{\theta}}_{ik}\right)\in {\mathbf{R}}^{d\times k}$ is the projection of ${\mathbf{X}}_{i}$ onto ${\mathbf{Q}}_{i}$. For the main geometric structure information within neighborhood, minimize the sum of square of distance between ${\mathbf{X}}_{i}$ and ${\mathbf{\Theta}}_{i}$:
where $\overline{{\mathbf{X}}_{i}}=1/k{\mathbf{X}}_{i}\mathbf{e}{\mathbf{e}}^{T}$ is a mean matrix of ${\mathbf{X}}_{i}$, $\mathbf{e}$ is a unit column matrix of length $k$, and ${\mathbf{I}}_{d}$ is a $d$ by $d$ unit matrix.
Apply singular value decomposition to ${\mathbf{X}}_{i}\overline{{\mathbf{X}}_{i}}$:
${\mathbf{Q}}_{i}{\mathbf{\Theta}}_{i}$, ${\mathbf{Q}}_{i}$, ${\mathbf{\Theta}}_{i}$ are computed as below:
where $\mathbf{U}\in {\mathbf{R}}^{D\times D}$ is a left singular vector set, $\mathbf{V}\in {\mathbf{R}}^{k\times k}$ is a right singular vector set, $\sum \in {\mathbf{R}}^{D\times k}$ is a singular value diagonal matrix, ${\sum}_{d}\in {\mathbf{R}}^{d\times d}$ is a diagonal matrix with $d$ maximum singular values in descending order, ${\mathbf{U}}_{d}\in {\mathbf{R}}^{D\times d}$ is the corresponding left singular vector set and ${\mathbf{V}}_{d}\in {\mathbf{R}}^{k\times d}$ is right singular vector set.
Thus, ${\mathbf{\Theta}}_{i}$ holds the most important geometric structure information in ${\mathbf{X}}_{i}$.
2.2. Global coordinate construction
Suppose ${\mathbf{T}}_{i}=\left({\tau}_{i1},{\tau}_{i2},\dots ,{\tau}_{ik}\right)\in {\mathbf{R}}^{d\times k}$ is the local lowdimensional coordinate of ${\mathbf{X}}_{i}$. Build the affine transformation between ${\mathbf{T}}_{i}$ and ${\mathbf{\Theta}}_{i}$:
The local reconstruction error ${\mathbf{E}}_{i}$ is written as:
where ${\mathbf{L}}_{i}\in {\mathbf{R}}^{d\times d}$ is a local affine matrix, $\overline{{\mathbf{T}}_{i}}=1/k{\mathbf{T}}_{i}\mathbf{e}{\mathbf{e}}^{T}$ is a mean matrix of ${\mathbf{T}}_{i}$, ${\mathbf{E}}_{i}\in {\mathbf{R}}^{d\times k}$ is a local reconstruction error matrix.
However, owing to the different contribution of samples to their manifold structure, a novel LCWR based on weight coefficient is proposed to reduce permutation error and reconstruct local coordinates more accurately. According to LCWR, the closer a sample is to its manifold, the larger its weight coefficient is. Likewise, the farther a sample is from its manifold, the smaller its weight coefficient is. So an exponential function of geodesic distance between a sample and the center point of its neighborhood, reflecting the proximity to its manifold structure, is defined as weight coefficient, namely:
where ${\omega}_{ij}$ denotes the weight coefficients of the $j$th nearest neighbor ${x}_{ij}$ in ${\mathbf{X}}_{i}$, ${G}_{ij}$ and ${\sigma}_{i}$ denote the geodesic distance from ${x}_{ij}$ to the center of ${\mathbf{X}}_{i}$ and the mean square error of ${\mathbf{X}}_{i}$ respectively, and $\beta $ is adjustment parameter.
Then ${\mathbf{E}}_{i}$ is rewritten as:
where ${\mathbf{W}}_{i}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sqrt{{\omega}_{i1}},\sqrt{{\omega}_{i2}},\cdots ,\sqrt{{\omega}_{ik}})\in {\mathbf{R}}^{k\times k}$is a weight coefficient matrix, ${\mathbf{H}}_{i}=(\mathbf{I}1/k\mathbf{e}{\mathbf{e}}^{T}){\mathbf{W}}_{i}\in {\mathbf{R}}^{k\times k}$.
Fix ${\mathbf{T}}_{i}$ and minimize ${\mathbf{E}}_{i}$ to preserve as much local information as possible, namely:
where ${\mathbf{L}}_{i}\in {\mathbf{R}}^{d\times d}$, ${\mathbf{\Phi}}_{i}={\mathbf{\Theta}}_{i}{\mathbf{W}}_{i}\in {\mathbf{R}}^{d\times k}$.
Substitute Eq. (9) into Eq. (7) and obtain ${\mathbf{E}}_{i}$:
where ${\mathbf{B}}_{i}={\mathbf{H}}_{i}{\mathbf{H}}_{i}{\mathbf{\Phi}}_{i}^{T}\times inv\left({\mathbf{\Phi}}_{i}{\mathbf{\Phi}}_{i}^{T}\right){\mathbf{\Phi}}_{i}\in {\mathbf{R}}^{k\times k}$.
Minimize the sum of all neighborhoods reconstruction errors to obtain the global lowdimensional coordinate $\mathbf{T}$:
where $\mathbf{T}=\left({\tau}_{1,}{\tau}_{2,}\dots ,{\tau}_{N}\right)\in {R}^{d\times N}$ is the global lowdimensional coordinate of $\mathbf{X}$, ${\mathbf{T}}_{i}=\mathbf{T}{\mathbf{R}}_{i}$, ${\mathbf{R}}_{i}=\mathbf{R}\left(:,{\mathbf{I}}_{i}\right)\in {\mathbf{R}}^{N\times k}$^{}is a selection matrix, $\mathbf{R}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\mathrm{o}\mathrm{n}\mathrm{e}\left(1,N\right)\right)$, ${\mathbf{I}}_{i}=\left({i}_{1},{i}_{1},\dots ,{i}_{k}\right)$ is the subscript of $k$ nearest neighbors of ${x}_{i}$.
It is equal to solve differential equation:
where $\mathbf{M}={\mathbf{M}}^{T}={\sum}_{i=1}^{N}{\mathbf{R}}_{i}{\mathbf{B}}_{i}{\mathbf{B}}_{i}^{T}{\mathbf{R}}_{i}^{T}\in {\mathbf{R}}^{N\times N}$.
Therefore, the optimal solution of ${\mathbf{T}}^{T}$ is composed of eigenvectors corresponding to the 2nd to the ($d+1$)th smallest eigenvalues of $\mathbf{M}$. ${\mathbf{T}}^{T}=\left({u}_{2},{u}_{3},\dots ,{u}_{d+1}\right)$ or $\mathbf{T}={\left({u}_{2},{u}_{3},\dots ,{u}_{d+1}\right)}^{T}\in {\mathbf{R}}^{d\times N}$ is an orthogonal global lowdimensional coordinate mapping matrix of nonlinear manifold in $\mathbf{X}$, where ${u}_{j}\in {\mathbf{R}}^{1\times N}$ is an eigenvector corresponding to the $j$th eigenvalue of $\mathbf{M}$.
LCWR is summarized as follows:
(1) Look for neighborhood. Local neighborhood ${\mathbf{X}}_{i}$ of each sample ${\mathbf{x}}_{i}$ ($i=\mathrm{1,2},\dots ,N$) is found by k nearest neighbors.
(2) Extract local coordinates. Projection matrix ${\mathbf{Q}}_{i}$ and local lowdimensional coordinate matrix ${\mathbf{\Theta}}_{i}$ of each neighborhood ${\mathbf{X}}_{i}$ are obtained according to Eq. (2) and Eq. (3), respectively.
(3) Construct global coordinates. Global lowdimensional coordinate matrix $\mathbf{T}$ is conducted from ${\mathbf{\Theta}}_{i}$ reconstruction by weight coefficient matrix ${\mathbf{W}}_{i}$ and expressed as Eq. (14).
The flow chart of LCWR is shown in Fig. 1.
3. Verification and analysis
Experimental data is from the bearing data center of Case Western Reserve University. Four types of bearing sates under speed of 1750 rpm and load of 1470 W, i.e. healthy state, outer race fault, inner race fault and ball fault with defective size of 0.3556 mm are considered. There are 48 samples for each state and total 192 samples for all states. Each sample acquired at 48 kHz includes 2048 points in length.
3.1. Highdimensional feature construction
As shown in Table 1, 12 timedomain statistical indicators and 8 frequencydomain statistical indicators are selected to constitute a 20dimensional sample to characterize the bearing state. An original sample signal is decomposed into eight subband signals by threelayer db8 wavelet packet decomposition, and the ratio of the energy of each subband to the total energy of all subbands is taken as the frequency domain indicator. That is, $e{}_{j}{}^{\mathrm{}}\mathrm{}={E}_{j}/E$, $E={\sum}_{j=1}^{8}{E}_{j}$, ${E}_{j}$ is the energy of subband signals. Thus, a highdimensional feature matrix $\mathbf{X}\in {\mathbf{R}}^{192\times 20}$ is created.
Fig. 1Procedure of LCWR
Fig. 2LCWR based fault diagnosis
Table 1Basic size and style requirements
Dimension  Description 
1  Standard deviation 
2  Variance 
3  Skewness 
4  Kurtosis 
5  Range 
6  Minimum 
7  Maximum 
8  Sum 
9  Root mean square 
10  Median 
11  Mean 
12  Crest factor 
1320  Energy ratio ${e}_{j}$ 
3.2. Lowdimensional feature extraction
According to Fig. 2, some of the lowdimensional feature samples extracted by LCWR are used as training samples to train support vector machine (SVM) while the others as test samples to be recognized by trained SVM. When using LCWR to extract bearing state characteristics, three parameters such as neighbor number $k$, dimension $d$ and adjustment parameter $\beta $ need to be optimized. Because the recognition rate can be regarded as a function of three parameters $k$, $d$ and $\beta $, these parameters interact to determine the recognition rate. By changing these parameters in a certain range and the corresponding recognition rate obtained, it is proved that there exist optimization values of parameters with the peak recognition rate. The trend of recognition rate with respect to a single parameter variable while the other two parameters fixed is shown in Fig. 35, respectively. Because of different parameters, the trend of recognition rate is also different from each other.
From the recognition rate about the nearest neighbor number $k$ in Fig. 3, there is the maximum rate 96.43 % at $k=$ 8. The role of $k$ on recognition rate is carried out by influencing the intrinsic geometry structure of highdimensional samples, the close relationship between similar samples and the nonlinear dimensionality reduction ability of LCWR. If $k$ is too small, LCWR can’t maintain the intrinsic geometry of highdimensional samples and close association between similar samples. If $k$ is too large, it weakens the nonlinear dimensionality reduction capability of LCWR. Hence, the lowdimensional manifold structure hiding in the highdimensional samples can be found to the greatest extent and achieve the maximum rate at the optimum value $k=$ 8. However, due to the comprehensive effect of different factors affected by $k$, the recognition rate fluctuates and there are multiple turnover points rather than a monotonous trend.
Dimension $d$ affects the recognition rate by mining the sensitive features of highdimensional samples in the neighborhood and eliminating redundant and interference components. From Fig. 4, it can be seen that the maximum recognition rate is 96.43 % at $d=$ 3, because an appropriate $d$ makes similar samples have approximate lowdimensional features, leading to better clustering effect and improvement in recognition rate. Otherwise, LCWR can’t fully mine the sensitive features from the highdimensional samples in neighborhood if $d$ is too small or the lowdimensional features contain redundancy and interference if $d$ is too large. Likewise, due to different factors, the recognition rate has multiple turnover points.
Adjustment parameter $\beta $ affects the recognition rate by changing the degree of clustering and global geometry retention. In relationship between $\beta $ and recognition rate in Fig. 5, if $\beta $ is too small, the proximity of samples is low and the clustering is obvious, but the degree of retention of the global geometric structure is poor. If $\beta $ is too large, the global geometric structure can be improved but the clustering reduced. These factors cause reduction in recognition rate. As a result, there is an optimal $\beta =$ 0.1 where the recognition rate reaches the maximum 96.43 %.
Fig. 3Recognition rate with neighbor k
Fig. 4Recognition rate with dimension d
Fig. 5Recognition rate with parameter β
The threedimensional sensitive features of bearing highdimensional characteristic samples reduced by LCWR according to the optimized values of $k=$ 8, $d=$ 3 and $\beta =$ 0.1 are shown in Fig. 6. After dimension reduction, four kinds of bearing samples have no intersection and overlap. Each of them has its own clustering center. The clustering effect and global geometric structure are objective, displaying a distinct manifold structure.
Fig. 6Embedded 3 dimensions by LCWR (k= 8, β= 0.1)
3.3. Dimensionality reduction effect analysis
LCWR is compared with LTSA, locally linear embedding (LLE) and principal component analysis (PCA) to verify its dimensionality reduction effect. The dimensionality reduction results of LTSA, LLE and PCA are shown in Fig. 7, Fig. 8, and Fig. 9, respectively. Generally speaking, the reduced dimensionality samples via these methods have different degrees of intersection and overlap, poor clustering within class, lack of clustering centers. It is difficult to mine the essential characteristics of the bearing state and the differences between classes. Although LTSA and LLE find the manifold structure of highdimensional samples, they are unable to expand the gaps between dissimilar samples in neighborhood. PCA belongs to one of linear statistical distributions without considering the local structure of the samples. It makes the intraclass aggregation poor and the differences between classes unclear, which fails to reveal the nonlinear manifold structure of the bearing state as shown in Fig. 9.
Fig. 7Embedded 3 dimensions by LTSA (k= 9)
Fig. 8Embedded 3 dimensions by LLE (k= 5)
Combining its weight coefficient with local coordinate’s permutation, LCWR enhances the intraclass aggregation and the differences between classes, overcomes the shortcomings that LTSA and LLE can’t enlarge the gaps between dissimilar samples in neighborhood, simplifies the dimension while retaining the lowdimensional principal characteristics of highdimensional samples, accurately reflects the relationship between signal characteristics and the bearing state, and effectively distinguishes four kinds of bearing states. As shown in Table 2, the features extracted by various methods are sent to SVM and the recognition rate of LCWR reaches the highest 96.43 % despite a little more time to run LCWR than LTSA, LLE and PCA as shown in Table 3. Therefore, in contrast to other dimensionality reduction methods, LCWR can achieve higher accuracy and prove its effectiveness.
Fig. 9Embedded 3 dimensions by PCA
Table 2Bearing condition recognition rate with various methods (%)
Sample  LCWR ($k=$8, $d=$3, $\beta =$0.1)  LTSA ($k=$9, $d=$3)  LLE ($k=$ 5, $d=$3)  PCA ($d=$3)  NDR ($d=$ 20) 
H vs. (O+I+B)  100  100  100  100  75 
O vs. (I+B)  93.06  93.06  95.83  97.22  66.67 
B vs. (O+I)  93.06  66.67  80.56  66.67  72.22 
I vs. (O+B)  97.22  95.83  87.50  94.44  95.83 
I vs. B  95.83  93.75  85.42  91.67  91.67 
O vs. I  100  95.83  93.75  100  95.83 
O vs. B  95.83  85.41  93.75  89.58  95.83 
Average rate  96.43  90.08  90.97  91.37  84.72 
Hhealthy, Oouter race defect, Iinner race defect, Bball defect, NDRnon dimensionality reduction 
Table 3Running time of SVM combined with various methods (s)
Sample  LCWR ($k=$8, $d=$3, $\beta =$0.1)  LTSA ($k=$9, $d=$3)  LLE ($k=$ 5, $d=$3)  PCA ($d=$3)  NDR ($d=$ 20) 
H vs. (O+I+B)  66  27  28  26  57 
O vs. (I+B)  73  27  31  26  78 
B vs. (O+I)  74  37  31  39  71 
I vs. (O+B)  69  26  26  27  58 
I vs. B  70  26  26  25  59 
O vs. I  62  24  26  24  56 
O vs. B  67  27  30  26  70 
Average time  71  28  28  28  64 
Besides, it can be founded that recognition rate of manifold dimensionality reduction using LCWR, LTSA, LLE and PCA (all greater than 90 %) is higher than that of the nondimensionality reduction method (only 84.72 %). It is further proved that these manifold learning methods can filter redundancy and interference of the highdimensional features and extract the intrinsic lowdimensional manifold characteristics, which can significantly improve the recognition rate of the bearing state as shown in Table 2. Meanwhile, these manifold learning methods except LCWR consume less time and get better recognition efficiency.
4. Conclusions
LCWR manifold learning is proposed to remove redundancy and noise in bearing highdimensional fault features and perform nonlinear dimensionality reduction for improvement in fault diagnosis capability. Geodesic distance based weight function is used to realign local coordinates to eliminate redundancy and interference in highdimensional feature samples and extract lowdimensional sensitive fault features. Experiments demonstrate that the intrinsic manifold structure of highdimensional feature samples can be well preserved after dimensionality reduction by LCWR, and the extracted lowdimensional feature samples can truly represent the nonlinear characteristics of different bearing states and the gaps between them. The lowdimensional feature samples are then identified by SVM, which results in a higher recognition rate than other methods. Thus, the effectiveness of LCWR is validated. In addition, LCWR is worth further studying to save running time.
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About this article
The work was supported by A Project Supported by Scientific Research Fund of Hunan Provincial Education Department (20A107), Project of National Natural Science Foundation of China (51875193) and Xiangtan guiding science and technology plan project (ZDXCG2019004). The authors thank all reviewers for their valuable comments and constructive advice on this paper.