## 3 Fault Feature Extraction Technique
### Sliding Matrix Sequences (SMS) Truncation and Sparse Reconstruction  
A vibration signal sampled by the sensor from rotating machinery is always a discrete time series with strong background noise. To research more detail of vibration signal, the original time series is separately divided into an array of short sliding signal segments by overlapping rectangle window.  
As seen in Figure 2, the sliding coefficient (\(s\)) is employed to an analyzed vibration signal \(y(t)\) (where \(t=1,2,\cdots N\)) with the length of \(N\). Firstly, the original signal is truncated into each signal segment and the \(i\)th short sliding signal segment \(Y_{i}\) is expressed as follows:  
\[Y_{i}=\left[y((i-1)s+1),y((i-1)s+2,\ldots y((i-1)s+Nw)\right]^{T} \tag{14}\]  
Then the SMS with \(z\) dimension is provided by the following form:  
\[\left\{\begin{array}{c}Y=\left[Y_{1},Y_{2}\ldots,Y_{z}\right]\\ z=(N-Nw)/s\end{array}\right. \tag{15}\]  
where \(z\) is the number of short sliding segment and \(Nw\) is the corresponding length.The scope of the sliding coefficient is defined as \(1<s<Nw\);  
To further extract the local feature information hidden in the SMS, sparse reconstruction by Hankel-matrix is utilized. Firstly, each column of the SMS is separately transformed into a block Hankel-matrix. Because a multi-dimensional matrix can help to deeply explore and analyze the short  
Figure 1: The diagram of the proposed feature extraction method.  
Figure 2: The diagram of sliding sequence truncation.  
sliding signal segment from different perspectives. \(A_{i}\) represents the \(i\)th block Hankel-matrix of the short time sliding signal \(\gamma_{i}\) and is expressed as  
\[A_{i}=M(Y_{i})_{(m\times n)}=\left[\begin{array}{cccc}y_{i}&y_{i+1}&\ldots&y _{i+n}\\ y_{i+1}&y_{i+2}&\ldots&y_{i+n+1}\\ \vdots&\vdots&\vdots&\vdots\\ y_{i+m}&y_{i+m+1}&\cdots&y_{i+m+n+1}\end{array}\right] \tag{16}\]  
where  
\[m+n+1=Nw \tag{17}\]  
Secondly, consider the Hankel-matrix as a graph and calculate the corresponding histogram of matrix data. According to the theory of image processing, the histogram can effectively exhibit the distribution of the pixel intensities in a predefined number of bins which range from the minimum to maximum intensity. On the other side, due to each row of the Hankel matrix being closely correlated, the intensity of data is sparsely distributed in the histogram. In other word, a sparse reconstructed sequence can be effectively achieved from Hankel-matrix by histogram. Then the sparse sequences of the \(i\)th Hankel-matrix \(A_{i}\) is presented as  
\[H_{i}=\left[h_{1,i},h_{2,i}\ldots,h_{2\;5\;6,i}\right]^{T} \tag{18}\]  
And the sparse reconstructed matrix sequence of the SMS can be written as  
\[H(Y)=\left[H_{1},H_{2}\ldots,H_{z}\right] \tag{19}\]  
As mentioned above, employ MDE_rms_ to each sparse reconstructed matrix sequence and achieve the corresponding characteristic index. The obtained calculation is called Multiscale Entropy Value Matrix (_MEVM_):  
\[MEVM=\left[E_{1},E_{2}\ldots E_{z}\right]=\left[\begin{array}{cccc}e_{1,1}&e_ {1,2}&\ldots&e_{1,z}\\ e_{2,1}&e_{2,2}&\ldots&e_{2,z}\\ \vdots&\vdots&\vdots&\vdots\\ e_{s,1}&e_{s,2}&\cdots&e_{s,z}\end{array}\right] \tag{20}\]  
where  
\[E_{i}=MDE_{RMS}(H_{i})=\left[e_{1,i},e_{2,i}\ldots,e_{z,i}\right]^{T} \tag{21}\]  
In the matrix, the column of \(MEVM\) is called Multiscale Entropy Value Sequences (MEVS) and represents the feature information of the same segment in different scales. And the row of \(MEVM\) separately represents the characteristic and complexity of different segments.  
According to Section 2.1, the \(MEVM\) is fused into a final \(MEEVS\) by LPP. The obtained \(MEEVS\) can be written as  
\[MEEVS=\left[\begin{array}{cccc}fe_{1}&fe_{2}&\ldots&fe_{z}\end{array}\right] \tag{22}\]