## 2 The Theory of the Smooth Local Subspace Projection Noise Reduction and PE
### Definition of PE  
Assuming a one dimensional time series \(\{x(i),i=1,2,\cdots,N\}\), and that it can be embedded into a high dimensional phase space by using phase space reconstruction, then the following matrix in Equation (16) can be obtained.  
\[\mathbf{X}=\left(\begin{array}{cccc}x(1)&x(1+\tau)&\cdots&x(1+(m-1)\tau)\\ x(2)&x(2+\tau)&\cdots&x(2+(m-1)\tau)\\ x(j)&x(j+\tau)&\cdots&x(j+(m-1)\tau)\\ &&\vdots\vdots\\ x(K)&x(K+\tau)&\cdots&x(K+(m-1)\tau)\end{array}\right) \tag{16}\]  
where \(j=1,2,\cdots,K\), \(m\) and \(\tau\) are embedding dimension and delay time respectively, \(K=n-(m-1)\tau\). Each row in the matrix \(\{x(j),\)\(x(j+\tau),\)\(\cdots,\)\(x(j+(m-1)\tau)\}\) can be regarded as a reconstructed component, and there is a total of \(K\) reconstructed components.  
In the reconstructed matrix \(\mathbf{X}\) is realigned in ascending order, and \(j_{1},\)\(j_{2},\)\(\cdots,\)\(j_{m}\) represent indices of each of the elements in the column of reconstructed components, which can be described in Equation (17).  
\[x(i+(j_{1}-1)\tau)\leq x(i+(j_{2}-1)\tau)\leq\cdots\leq x(i+(j_{m}-1)\tau) \tag{17}\]  
If there are equivalent values in reconstructed components, then the elements should be aligned according to the values of \(j_{p}\) and \(j_{q}\). When there are two of equal value, and \(j_{p}<j_{q}\) in the following situation, it can be ranked as in Equation (18) in order to solve the same value situation.  
\[x(i+(j_{p}-1)\tau)\leq x(i+(j_{q}-1)\tau) \tag{18}\]  
It can be obtained from above that, for any time series, each row in the reconstructed matrix \(\mathbf{X}\) can generate a set of symbolic sequence, as shown in Equation (19).  
\[s(l)=\{j_{1},j_{2},\ldots,j_{m}\} \tag{19}\]  
where \(l=1,2,\cdots,k\), and \(k\leq m!\). There is a total of \(m!\) kinds of different mapping symbol sequences \((j_{1},\)\(j_{2},\)\(\cdots,\)\(j_{m})\) of \(m\) dimensional phase space, and symbol sequence \(s(l)\) is one of them. The probability of each symbol sequence is \(P_{1},\)\(P_{2},\cdots,\)\(P_{k}\) respectively, the PE [35]\(H_{p}\) of \(k\) kinds of different symbol sequences of a time series \(x(i)\) is defined according to information entropy, as shown in Equation (20).  
\[H_{p}(d)=-\sum_{j=1}^{k}P_{j}\text{In}(P_{j}) \tag{20}\]  
The size of \(H_{p}(d)\) value represents the random degree of the time series: The smaller value of \(H_{p}(d)\) means the more regular of the time series, and the larger value of \(H_{p}(d)\) means the more random of the time series. The change of \(H_{p}(d)\) can show and enlarge the detail changes of time series. In gear and bearing fault diagnosis, the more serious the fault type is, the more regular of the one dimensional time series is, the smaller PE will be.  
In this paper, a novel method of applying PE in the fault classification of rotary machinery based on the smooth local subspace projection denoising method is proposed. After experimental studies, it can be obtained that the smooth projection denoising method has an obvious noise reduction effect towards a nonlinear and nonstationary signal. Thus, the smooth local subspace projection method can be well applied to a variety of mechanical fault signals. As for gear and bearing parts, different faulttypes correspond to different fault degrees. The smaller entropy means the less complexity of fault signal type, namely that when the fault type is more serious, the fault signal has better predictability, and its entropy will be smaller. From all above theoretical derivations, the smooth local subspace projection method can be used to denoise gear and bearing fault signals, and PE is utilized to the classification of different faults. The basic idea of the proposed method is illustrated in Figure 7.