## 3 Proposed Fault Diagnosis Method
### Proposed TfEn-Based Feature Extraction  
Entropy is the quantitative tool to describe the complexity and irregularity [24] of systems, which varies with the changing status of systems. For the centrifugal pumps, if the signal is approximately stationary (such as the first 5 Imfs shown in Figure 2), we usually think that the distribution of signals in the frequency domain better reflect the status of objects, where the existing entropy aiming at 1D sequence may be enough to measure their complexity. While, if the signal is close to non-stationary (such as the last 5 Imfs shown in Figure 2), both the time domain and frequency domain contain the useful information reflecting the status of objects. In this case, the existing entropy cannot comprehensively quantify the irregularity of signals in two dimensions, which inevitably bring about the loss of useful information. Aiming at addressing the above-mentioned issues, we propose TfEn to jointly measure the complexity and dynamic change though taking into account the nonlinear behaviors from both dimensions of the time and frequency. The process of TfEn is introduced as follows. 1. Given a 1D time series \(x(t)\). For the purpose of measuring its complexity both in time and frequency domain, \(x(t)\) should be first extended into the 2D time-frequency matrix. In this paper, short-time Fourier transform (STFT) [25], a classical time-frequency analysis method, is applied to achieve the above target considering its balance of calculation efficiency and performance. \[X(t,f)=\int_{-\infty}^{\infty}x(\tau)w(\tau-t)e^{-j2\pi f\tau}\mathrm{d}\tau,\] (6)  
Figure 2: 10 sub-signals decomposed by complementary ensemble empirical mode decomposition (CEEMD). where \(w(t)\) is the window function, \(t\) and \(\tau\) are both the time variables and \(f\) is the frequency variable. \(X(t,f)\) can be illustrated as the correlation between \(x(\tau)\) and \(w(\tau-t)e^{-j2\pi f\tau}\). 2. Further, through the square operation, the \(X(t,f)\) will be transformed into the energy matrix. \[M=|X(t,f)|^{2}=\left[ \begin{array}{cccc}e_{11}&e_{12}&\cdots&e_{1Y}\\ e_{21}&e_{22}&\cdots&e_{2Y}\\ \vdots&\vdots&e_{ij}&\vdots\\ e_{X1}&e_{X2}&\cdots&e_{XY}\end{array} \right],i=1,\ldots,X,j=1,\ldots,Y,\] (7) where the \(M\) is a 2D matrix consisting of \(X\times Y\) elements \(e_{ij}\), and \(e_{ij}\) can be interpreted as the energy located at frequency \(i\) and time \(j\). 3. According to the [PERSON] methodology, the average degree of uncertainty depicts the irregularity of probability system. In this case, to obtain the robust representation of the time-frequency energy matrix \(M\), we utilize a sliding window of size \(F\times T\) to divide the \(M\) into \(I\times J\) blocks \(W_{i,j}\); \(i=1,\ldots,I,j=1,\ldots,J\). Then a normalization operator will process these blocks one by one and transform them into a probability density matrix \(Q\). The above operation can be illustrated in Figure 3.  
As shown in Figure 3, the \(F\), \(T\), \(L_{f}\) and \(L_{t}\) are 4 hyper-parameters of our proposed Tfen, which respectively represent the span of frequency-axis, span of time-axis, translation span of frequency-axis and translation span of time-axis of each time-frequency block. Figure 3: Matrix division and normalization operations using time-frequency entropy (TfEn). As shown in Figure 3, through the normalization operators, each time-frequency energy block \(W_{i,j}\) will be transformed into a probability density reflecting its irregularity. The calculation of the normalization operators is described as follows:  
\[w_{i,j}=\sum_{y=1+L_{i}\times j}^{T+L_{i}\times j}\sum_{x=1+L_{j}\times i}^{F+L_{ j}\times i}e_{x,y}, \tag{8}\]  
\[A=\sum_{i=1}^{I}\sum_{j=1}^{J}w_{i,j}, \tag{9}\]  
\[q_{i,j}=\frac{w_{i,j}}{A}, \tag{10}\]  
where \(e_{x,y}\) is an energy element of the matrix \(M\) at coordinate \((x,y)\), \(w_{i,j}\) is defined as the sum of energy values in the block \(W_{i,j}\), and \(A\) is defined as the total sum of energy elements of all blocks.