Time-Frequency Fault Feature Extraction for Rolling Bearing Based on the Tensor Manifold Method

Rolling-bearing faults can be effectively reflected using time-frequency characteristics. However, there are inevitable interference and redundancy components in the conventional time-frequency characteristics. Therefore, it is critical to extract the sensitive parameters that reflect the rolling-bearing state from the time-frequency characteristics to accurately classify rolling-bearing faults. Thus, a new tensor manifold method is proposed. First, we apply the Hilbert-Huang transform (HHT) to rolling-bearing vibration signals to obtain the HHT time-frequency spectrum, which can be transformed into the HHT time-frequency energy histogram. Then, the tensor manifold time-frequency energy histogram is extracted from the traditional HHT time-frequency spectrum using the tensor manifold method. Five time-frequency characteristic parameters are defined to quantitatively depict the failure characteristics. Finally, the tensor manifold time-frequency characteristic parameters and probabilistic neural network (PNN) are combined to effectively classify the rolling-bearing failure samples. Engineering data are used to validate the proposed method. Compared with traditional HHT time-frequency characteristic parameters, the information redundancy of the time-frequency characteristics is greatly reduced using the tensor manifold time-frequency characteristic parameters and different rolling-bearing fault states are more effectively distinguished when combined with the PNN.


Introduction
Rolling bearings are widely used in modern rotating machinery, and their failure is one of the most common causes of machine breakdowns and accidents [1][2][3].Therefore, fault diagnosis of rolling bearings is necessary to ensure the safe and efficient operation of machines in engineering applications.The main aspects of bearing fault diagnosis are classification and pattern recognition, where feature extraction directly affects the accuracy and reliability of the fault diagnosis [4].Rolling-bearing fault features can be generally divided into three categories: time-domain characteristics, frequency-domain characteristics, and time-and frequencydomain characteristics [5,6].
Time-domain characteristics are fairly intuitive; however, they fluctuate significantly and lack quantitative judging criteria.Thus, they cannot be directly used to diagnose bearing faults.In contrast, frequency-domain characteristics can be used to diagnose bearing fault conditions more accurately because different bearing faults correspond to different characteristic frequencies.However, there are typically noise and modulation components in the bearing fault signals.Thus, direct application of the frequency-domain method will submerge the fault characteristic frequency in noise or false frequency components because of improper selection of the demodulation parameters.Furthermore, signal denoising and demodulation must be conducted before extracting the bearing fault characteristic frequencies.In the process of signal denoising and demodulation, parameters such as the denoising parameters, demodulation center, and filter bandwidth should be properly selected based on experience, and a satisfactory selection is only obtained after numerous adjustments.
The time-and frequency-domain characteristics, which have the intuitive feature of the time-domain characteristics 2 Mathematical Problems in Engineering and good time-frequency aggregation, can simultaneously reflect the time-domain and frequency-domain characteristics of a signal [7][8][9][10].Therefore, extracting the timefrequency fault characteristics is important for fault diagnosis.Wang and Hu used the principle of time-frequency image analysis to diagnose gearbox faults in 1993 [11]; this effort was the first application of time-frequency image for the fault diagnosis of machinery and equipment.Zhang et al. subsequently used time-frequency images to classify diesel engine faults under complex vibration conditions [12].Zhu et al. used short-time Fourier transform to extract timefrequency features for fault diagnosis [13], and satisfactory results were achieved.However, the aforementioned timefrequency characteristics are not adaptive and can only be used for reciprocating machinery.To overcome the limitations of the above methods, Huang et al. proposed the HHT time-frequency spectrum, which is self-adaptive [14].The HHT time-frequency spectrum is suitable for analyzing nonstationary signals because of its frequency instantaneity [15].However, mode mixing is inevitable for signals with instantaneous frequency trajectory crossings [16,17].Li et al. used the geometric center of the HHT time-frequency spectrum as a feature vector [18,19] in combination with SVM and classified rolling-bearing fault signals.However, because the geometric center requires a considerable amount of calculations and lacks corresponding physical meaning, it can only provide qualitative classification criteria.Manifold learning has recently emerged in nonlinear-feature extraction because of its capability of effectively identifying hidden low-dimensional nonlinear structures in high-dimensional data.He [20] proposed a time-frequency manifold feature by combining the time-frequency distribution and the nonlinear manifold for an effective quantitative representation of machinery health pattern.
This paper proposes a new tensor manifold timefrequency feature extraction method to overcome the weakness of traditional HHT time-frequency characteristics.The HHT time-frequency spectrum, which contains a considerable amount of failure information, is used as the research object.The tensor manifold learning method is applied to extract the tensor manifold time-frequency characteristics of the HHT time-frequency spectrum.The two-dimensional time-frequency information does not need to be converted into a one-dimensional vector when calculating the tensor manifold, and the information loss is significantly reduced.On this basis, five time-frequency characteristic parameters are defined.The tensor manifold time-frequency characteristic parameters can distinguish different rolling-bearing fault states more effectively than traditional HHT timefrequency characteristic parameters.Combined with PNN, the tensor manifold time-frequency characteristic parameters can effectively distinguish different rolling-bearing fault states.Engineering vibration signals were used to evaluate the efficiency of the proposed method.
The remainder of this paper is organized as follows.The theory basis is introduced in Section 2, and the tensor manifold time-frequency fault feature extraction method is described in Section 3. Section 4 presents the adaption of the proposed method to rolling-bearing fault classification.Rolling-bearing fault classification is implemented in Section 5. Finally, conclusions are drawn in Section 6.

Theory Basis
2.1.HHT Time-Frequency Spectrum.Based on the definition of instantaneous frequency and EMD, the HHT timefrequency spectrum is analytically derived as follows.
Apply the EMD to signal () to obtain the IMFs of ().Then, the analytical form of () can be expressed as where Re is the real part of the selected signal,   () is the instantaneous amplitude of the ith IMF, and   () is the corresponding instantaneous frequency.The time, frequency, and amplitude of the signals can be combined to form the three-dimensional time-frequency space.Then, the amplitude distribution on time-frequency plane is referred to as the HHT time-frequency spectrum, which is expressed as where Re is the real part of the selected signal and   is the indicator variable.When   = ,   = 1, and when   ̸ = ,   = 0.
The HHT time-frequency analysis is a decomposition method based on signal local characteristics, which provides a physical basis for the concept of instantaneous frequency and sets this method apart from conventional methods through its use of numerous harmonic components to describe complex nonlinear and nonstationary signals.Therefore, from the concept definition and the nature of signal analysis, the HHT time-frequency spectrum eliminates the limitations of Fourier transform and can accurately describe nonstationary signal characteristics.

Locality Preserving Projection (LPP) Manifold Learning
Algorithm.The LPP manifold learning algorithm aims at finding the linear transformation matrix W to reduce the dimensionality of high-dimensional data.There are  training samples {x  }  =1 ∈ R  , and W can be obtained by minimizing the following objective function: where  , is the similarity measure among objects and can be defined using the k-nearest-neighbor method: where (, x  ) denotes the  nearest neighbor of x  and  is a positive constant.Both  and  can be determined empirically.Equation (3) demonstrates the feature space after dimension reduction can maintain the local structure of the original high-dimensional space.We apply an algebraic transformation to (3) as follows: where X = [x 1 , x 2 , . . ., x  ], D denotes an  ×  diagonal matrix, where the diagonal element   = ∑    , S = (  ) × , and Then, the problem of solving for the optimal vector W can be transformed into the following eigenvalue problem:  [22] to extract the time-frequency spectrum fault characteristics.
There are  two-dimensional training images {A  }  =1 ∈ R × , where  denotes an -dimensional unitization column vector.The main objective of the tensor manifold algorithm is to make each  ×  image matrix A  project onto  using a linear transformation y  = A  .In this manner, an dimensional column vector can be obtained and considered a projection feature vector of image A  .The objective function of the tensor manifold algorithm is expressed as follows: where A = [A 1 , A 2 , . . ., A l ], the definitions of D and L are identical to those in the LPP manifold learning method, and ⊗ denotes the Kronecker product.
Then, the problem of solving for the optimal vector  is transformed into the following eigenvalue problem: where  is comprised of  feature vectors that correspond to the smallest nonzero eigenvalues; that is, there are  optimal projection vectors , which can form the projection matrix W = [ 1 ,  2 , . . .,   ].For any image A  , there is where y 1 , y 2 , . . ., y  are the projection feature vectors of the sample image A  and y  = [y 1 , y 2 , . . ., y  ], which is comprised of projection feature vectors, is the characteristic matrix of the sample image A  .

Probabilistic Neural Network (PNN).
The neural composition structure and elements of the PNN are shown in Figure 1 In the PNN, characteristic parameters were transported into each node on the pattern layer through the input layer.Then, we apply layer nonlinear mapping to the input parameters in each node of the PNN pattern and complete the comparison between an unknown type with a known type.Finally, the characteristic parameters that represent the types are input to the next layer for processing.The node structure of the layers is used to be called the RBF center, and the node output is expressed as follows: where the th center vector is   , which is the same size as the input vector.  (⋅) denotes the radial basis function, which is typically a Gaussian function; that is, where   denotes the shape parameter that corresponds to the th component of the radial basis function.
To facilitate the calculation, X and   are processed with mathematical regularization and unit.Assume that z  = X⋅  .Then, the above expression is expressed as follows: Finally, through the output layer (decision-making layer), the characteristic parameters, which are derived from the pattern layer, are accumulated to provide the category feature vector, which is The PNN has the following characteristics: (1) the training convergence speed is high, making the PNN suitable for the real-time processing of various data types; (2) the pattern unit can form any nonlinear mapping judgment surface, which is closest to the optimal judgment surface bayes; (3) the selection of the RBF center kernel function has diversity, and the form of the kernel function has a small effect on the recognition results; and (4) the number of neuron nodes in each PNN layer is relatively stable, the hardware processing is convenient, and the fault tolerance is high.The PNN has been widely used in pattern recognition, prediction estimation, and filtering denoising.(a)

Tensor Manifold Time-Frequency Fault
Feature Extraction Method 3.1.Description of the Proposed Method.The manifold learning method is a nonlinear dimension reduction method to extract low-dimensional nonlinear characteristics from highdimensional data.Unlike the conventional linear dimension reduction methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and linear discriminant analysis (LDA), this method is a nonlinear method to address the part before the whole.By satisfying the entire optimization, the manifold learning method can preserve the partial manifold characteristics and effectively extract the nonlinear manifold characteristics that are inherent in the high-dimensional characteristic set.However, the manifold learning algorithm suffers from information loss and error that are caused by the transformation from a set of two-dimensional time-frequency characteristics to a onedimensional vector.
To alleviate the drawback of information loss and error, this section presents a tensor manifold time-frequency fault feature extraction method based on the tensor manifold algorithm to extract the set of low-dimensional timefrequency characteristics from the set of high-dimensional time-frequency characteristics.Then, five tensor manifold time-frequency characteristic parameters were defined and combined with the PNN to classify the rolling-bearing failure samples.
The tensor manifold time-frequency fault feature extraction method is described as follows, and Figure 2 presents its flow chart.
(1) Group the rolling-bearing vibration signal samples to be classified and for training and then calculate the HHT time-frequency spectrum.To hasten the calculation of the tensor manifold algorithm, grid the time-frequency regions, integrate the energy value of the HHT time-frequency spectrum of each mesh, and convert the HHT time-frequency spectrum into HHT time-frequency energy histograms.Input the defined parameters of the training signal samples into the PNN for the rolling-bearing fault classification.
(5) Input the tensor manifold time-frequency characteristic parameters of the to-be-classified signal samples into the trained PNN to classify the rolling-bearing faults.

Definition of the Time-Frequency Characteristic Parameters.
The tensor manifold time-frequency energy histogram is a nonlinear time-frequency fault feature and can effectively differentiate different rolling-bearing fault signals.However, it is equal to a two-dimensional matrix, which makes it unsuitable for direct application in fault classification.In this section, several parameters are presented to quantitatively measure the difference among the tensor manifold timefrequency energy histograms.Their definitions are provided as follows.

Energy Entropy.
Entropy is proposed to measure the data complexity and the probability to generate the new signal model.Here, the energy entropy is defined as follows: where  is the energy entropy,   is the value of the time frequency energy histogram, and   is the proportion of each   in the total ∑  =1   .In addition, the energy entropy can reflect the uncertainty in the energy distribution.

Energy Correlation
Because each E  varies with different time-frequency energy histograms, we can analyze the relevance of E  and E  to measure the difference in the time-frequency energy histogram.The energy correlation coefficient vector is defined as follows: where corcoef() = corcoef(E  , E  ) and corcoef(⋅) is the cross-correlation function.

Energy Sparsity.
The signal energy distribution of the time-frequency energy histogram varies more significantly as it approaches zero.The sparsity expresses the sparse distribution of energy, and the purpose of estimating the sparsity is to obtain a function (x), x ∈ R  .If x is sparse, then (x) is relatively large, and vice versa.Generally, the norm   of vector x is used to quantitatively estimate the sparsity.Here, we define the   norm of the standardized form of vector x as follows: where 1 ≤  < ∞, and we select  = 1 such that  1 can accurately reflect the energy distribution of the histogram.

Energy Mutual Information.
Mutual information is proposed to measure the degree of independence among random variables.The mutual information of multiple variables is defined as the  divergence of the multivariate joint probability density and its marginal probability density product: where  is the divergence,  =  1 ,  2 , . . .  , () is the multivariate joint probability density function, and   (  ), ( = 1 ∼ ) is the marginal probability density function.Then  is defined as follows: where (), () are two different probability density functions of a random vector x.The energy mutual information can be calculated for each histogram according to (18).The test rig, which is shown in Figure 3, was constructed for the run-to-failure testing of the rolling bearing.A 1.5 kW Although there are several differences among the four types of signals in the time-domain wave nature, it is difficult to distinguish the rolling-bearing fault conditions using these intuitive qualitative differences.Therefore, the fault features that quantitatively represent the differences of different rolling-bearing fault statuses must be studied.

HHT Time-Frequency Characteristics of Rolling Bearings.
First, the HHT time-frequency spectra of four states were calculated using the HHT method.Then, the HHT timefrequency spectrum was divided into 64 regions of identical size.The histogram of the HHT time-frequency spectrum was obtained via the integral of the energy amplitude for each region.Different types of signal HHT time-frequency spectra and their histograms are shown in Figure 5.
Figure 5 describes the HHT time-frequency spectrum and corresponding time-frequency energy histograms of the normal rolling-bearing vibration signal.As shown in Figure 5, the time-frequency energy is mainly distributed in the low-frequency region and decreases with increasing frequency.The amplitude ranges from 0 to 20 g 2 .Figure 6 presents the HHT time-frequency spectrum and corresponding time-frequency energy histogram with an inner-race fault.As shown in Figure 6, the time-frequency energy is widely distributed in both the low-and highfrequency regions, with a lull in the mid-frequency region.The amplitude ranges from 0 to 40 g 2 , the maximum value of which is greater than that in the normal case.
Figure 7 presents the HHT time-frequency spectrum and corresponding time-frequency energy histograms with a ball fault.As shown in Figure 7, the time-frequency energy is mainly distributed in the high-frequency region and exhibits a less significant yet stable distribution in the low-frequency region.The amplitude of the energy histogram ranges from 0 to 30 g 2 .Figure 8 presents the HHT time-frequency spectrum and corresponding time-frequency energy histograms with an outer-race fault.As shown in Figure 8, the time-frequency energy is centered in the high-frequency region and exhibits a lull in the low-frequency region.The distribution trend begins at a rather low frequency and increases abruptly at a certain high frequency.The magnitude ranges from 0 to 100 g 2 .the energy entropy shown in Figure 9(b) cannot provide a clear distinction between inner-race faults and ball faults.Thus, the manifold energy entropy is more appropriate for classifying rolling-bearing faults.

Manifold Energy Correlation Coefficient. The manifold energy correlation coefficient (manifold energy corcoef(𝑖)
for short) is obtained by calculating the manifold energy corcoef() between E 1 , E 2 , . . ., E 6 and E  .The results are shown in Figure 10.
As shown in Figure 10, the manifold energy corcoef() can generally distinguish different fault signals, but different manifold energies corcoef() have different abilities.First, corcoef(1) can generally distinguish four rolling-bearing failures.corcoef(2) is also suitable for distinguishing failures, except for the normal and ball fault samples.corcoef(3) failed to distinguish the ball fault and outer-race fault, corcoef(4) failed to distinguish the inner-race fault and ball fault, and corcoef( 5) and corcoef (6) failed to distinguish all faults.Thus, corcoef(1) is accepted as the parameter that is best able to distinguish different rolling-bearing failures.
Figure 11 presents the energy correlation coefficient (hereafter denoted as "energy corcoef()"), which is calculated using six large energy bands of the HHT time-frequency energy histogram without manifold analysis.As shown in Figure 11, the energy corcoef(), where  = 1, . . ., 6, cannot provide clear distinctions and thus is not suitable for classifying different rolling-bearing faults.

Manifold Energy Sparsity.
The energy distributions of different fault signals are different, as are the energy distributions of different regions in the time-frequency energy histogram.Figure 12(a) presents the manifold energy sparsity of four rolling-bearing signals.As shown in Figure 12(a), the manifold energy sparsity can effectively distinguish different fault samples and can be used to classify different rollingbearing faults.Figure 12(b) presents the energy sparsity of four types of signal samples.Although the energy sparsity can distinguish different rolling-bearing samples, the energy sparsity within the sample fluctuations, and the difference in energy sparsity of the inter-class sample is not obvious.Therefore, the energy sparsity is not a rolling-bearing fault parameter.samples into 6 regions based on the frequency.Then, we calculate the corresponding mutual manifold energy information and mutual energy information.As described in Figure 13(a), different rolling-bearing faults can be accurately distinguished using the mutual manifold energy information, which is clearly different among the fault samples; thus, the mutual manifold energy information can be used as the rolling-bearing fault characteristic parameter.Figure 13(b) illustrates that the mutual energy information of normal and ball fault samples is similar, and, thus, these two fault types cannot be distinguished.Therefore, the energy mutual information is not suitable for use as the rolling-bearing fault characteristic parameter.

Manifold Energy Kurtosis.
We calculate the manifold energy kurtosis and energy kurtosis based on the corresponding energy histograms.Figure 14( energy kurtosis of the four different samples; the manifold energy kurtosis varies significantly, and, thus, the mutual energy manifold information can be used to classify different rolling-bearing faults.Figure 14(b) presents the energy kurtosis of the four different samples; as shown, all samples exhibit highly similar energy kurtosis values.In particular, the values of the normal fault, ball fault, and outer-race fault are extremely similar.Therefore, we cannot distinguish different rolling-bearing faults using energy kurtosis.features (e.g., subband energy), and time-frequency domain features (e.g., HHT time-frequency spectrum).These simple features are not as advantageous as the tensor manifold timefrequency features for capturing synthetic signal information.Thus, we use the tensor manifold time-frequency parameters for rolling-bearing fault diagnosis in this paper.
To demonstrate the benefit of the proposed parameters, simple features based on the HHT time-frequency spectrum are also conducted to analyze the four types of rolling-bearing fault signals.The test results are presented in Figures 9-13.These simple features perform worse in classification than do the tensor-manifold-based features.To avoid possible mistakes in pattern identification, it is necessary to improve the classification capability for reliable pattern diagnosis by exploring advanced features, which is the purpose of this paper.

Rolling-Bearing Fault Classification
The preceding analysis demonstrates that the five parameters (manifold energy entropy, manifold energy correlation coefficient, manifold energy sparsity, manifold energy mutual information, and manifold energy kurtosis) can efficiently distinguish the rolling-bearing fault states.Thus, they are used as the PNN input parameters for the bearing fault classification.
To verify the effectiveness of the manifold feature for identifying the four bearing faults, 20 samples of four types (normal, inner-race faults, ball faults, and outer-race faults) were used as training samples.The other 20 samples of each type were used for classification purposes.Each sample was extracted for the five aforementioned manifold feature parameters.The characteristic parameters of 80 training samples were used to train the PNN, and the numbers of nodes in the four PNNs were 5, 30, 4, and 4. Finally, the characteristic parameters of the 80 to-be-classified samples were input into the PNN for classification.The PNN classification results of the four bearing faults are shown in Table 1.
Table 1 illustrates that, when the tensor manifold timefrequency characteristic parameters are used as inputs for the PNN, four types of rolling-bearing fault samples can be effectively distinguished and each of the 20 to-be-classified samples for each type of fault can be correctly classified.The normal sample classification exhibits the best results, whereas the minimum components of the category vectors of the inner-race fault samples, ball fault samples, and outerrace fault samples are 0.92, 0.93, and 0.92, respectively.The classification results of the PNN indicate that the rollingbearing fault condition can be effectively described using the tensor manifold time-frequency characteristic parameters and that the rolling-bearing fault type can be accurately identified with the PNN.
To compare the proposed method with traditional extraction methods, we extract the five defined parameters of the same training and to-be-classified samples using the traditional HHT time-frequency method as the PNN input parameters for the bearing fault classification.The results are shown in Table 2.
Table 2 illustrates that, when the HHT time-frequency characteristic parameters are used as inputs to the PNN, four types of rolling-bearing fault samples can generally be distinguished, but the distinction is not adequate.The normal sample classification exhibits the best results, whereas the minimum components of the category vector of the innerrace fault samples, ball fault samples, and outer-race faults are 0.69, 0.76, and 0.73, respectively.
In Table 1, the minimum components of the category vectors of the four bearing faults are 0.92.In contrast, in Table 2, except the normal-state, the components of the  category vectors of the inner-race fault, the ball fault, and the outer-race fault above 0.85 account for 65%, 80%, and 60%, respectively.Compared with the results of the PNN, which uses tensor manifold time-frequency characteristic parameters as inputs, the classification performance with traditional HHT time-frequency features is relatively poor.
To verify the effectiveness of the manifold feature for identifying different damage degrees in the same fault type of rolling-bearing status, the proposed method was used to classify the inner-race fault samples with different damage degrees.20 samples of four degrees (normal, mild damage, moderate damage, and severe damage) were used as training samples.The other 10 samples of each degree were used for classification purposes.The PNN classification results of the to-be-classified inner-race fault samples are shown in Table 3.
To compare the proposed method with traditional extraction methods, we extract the five defined parameters of the same training and to-be-classified inner-race fault samples with different damage degrees using the traditional HHT time-frequency method as the PNN input parameters for the bearing fault classification.The results are shown in Table 4.
Table 4 depicts that, when the HHT time-frequency characteristic parameters are used as inputs to the PNN, inner-race fault samples with different damage degrees can generally be distinguished, but the distinction is not adequate.The minimum components of the category vector of the inner-race mild-damage fault samples, moderate-damage fault samples, and severe-damage fault samples are 0.83, 0.86, and 0.78, respectively.
In Table 3, the minimum components of the category vectors of the four bearing faults are 0.93.In contrast, in Table 4, except the normal-state, the components of the category vectors of the inner-race fault, the ball fault, and the outer-race fault above 0.85 account for 90%, 100%, and 60%, respectively.Compared with the results in Table 3, the results of the PNN, which uses the traditional HHT time-frequency characteristic parameters as inputs, indicate a relatively poor classification performance.

Conclusion
This paper studies the problem of rolling-bearing fault feature extraction.A time-frequency feature extraction method based on tensor manifolds for rolling bearings was proposed to overcome the deficiencies of the traditional HHT timefrequency feature extraction methods and to remove redundant time-frequency feature information.The HHT timefrequency energy histograms of the rolling-bearing fault signal were used to compose high-dimensional time-frequency fault feature sets.On this basis, the signal time-frequency characteristics were extracted using tensor manifold learning.Five tensor manifold time-frequency characteristic parameters were defined: manifold energy entropy, manifold energy correlation coefficient, manifold energy sparsity, manifold energy mutual information, and manifold energy kurtosis.These characteristic parameters and a PNN were combined to accurately classify rolling-bearing fault samples.The tensor manifold method can realize the nonlinear fusion of the time-frequency information, which can effectively extract the intrinsic nonlinear characteristics of high-dimensional timefrequency combination, and avoid the loss of information

Figure 1 :
Figure 1: Internal composition structure and elements of the PNN: (a) system structure of the PNN, (b) pattern unit of the PNN, and (c) output unit of the PNN.

( 2 )
The HHT time-frequency energy histograms are essentially two-dimensional matrices.Use the HHT time-frequency energy histograms that correspond to signal samples to form a set of high-dimensional time-frequency characteristics.(3)Apply the tensor manifold algorithm to extract the set of low-dimensional time-frequency characteristics from the set of high-dimensional time-frequency characteristics.In this manner, the tensor manifold time-frequency energy histograms are obtained.(4) Based on the result of step (3), define different tensor manifold time-frequency characteristic parameters.

Figure 2 :
Figure 2: Flow chart of the time-frequency characteristic extraction method based on a tensor manifold.

Figure 3 :
Figure 3: Diagram of the experimental test rig.

Figure 6 :Figure 7 :Figure 8 :Figure 9 :
Figure 6: (a) HHT time-frequency spectrum with an inner-race fault and (b) time-frequency energy histogram with an inner-race fault.

Figure 10 :
Figure 10: Manifold energy correlation coefficient of the four fault signals.

Figure 11 :
Figure 11: Energy correlation coefficient of the four fault signals.

4. 4 .Figure 12 :Figure 13 :
Figure 12: (a) Manifold energy sparsity of the four fault signals and (b) energy sparsity of the four fault signals.

Figure 14 :
Figure 14: (a) Manifold energy kurtosis of the four fault signals and (b) energy kurtosis of the four fault signals.
Coefficient.Divide the timefrequency energy histogram into  sections by frequency and mark E 1 , E 2 , . . ., E  and E The energy values are mainly within the middle range, and larger or smaller values are less likely to occur.For the sequence X =  1 ,  2 , . . .  of energy values, the overall kurtosis is defined as [23]osis.Kurtosis is a physical parameter that is proposed to measure the degree of Gaussian distribution of a random variable.A larger energy kurtosis in the time-frequency energy histogram corresponds to weaker Gaussianity of the energy distribution, whereas a smaller kurtosis indicates stronger Gaussianity.If the Gaussianity of the energy distribution is strong, the energy distribution presents the "middle big, two sides small" phenomenon.4.1.Rolling-Bearing Fault Data.To verify the effectiveness of the proposed method, the new method was used to analyze bearing fault data from the Bearing Data Center of Case Western Reserve University[23].

Table 1 :
Classification results of the four bearing faults using the PNN with the proposed tensor manifold features.

Table 2 :
Classification results of the four bearing faults using the PNN with traditional HHT time-frequency features.

Table 3 :
Classification results of the inner-race fault using the PNN with the proposed tensor manifold features.

Table 4 :
Classification results of the inner-race fault using the PNN with traditional HHT time-frequency features.Compared with the HHT time-frequency characteristic parameters, the tensor manifold time-frequency characteristic parameters can more effectively distinguish the four bearing faults, different damage degrees in the same fault type, and different fault types with different damage degrees because of its strong nonlinearity and reduced information redundancy.The effectiveness of the proposed method was verified using real rolling-bearing fault signals.Thus, this paper provides an important method to solve the rolling-bearing feature extraction problems.