## 3 Proposed Method
### The Hybrid Networks
#### 3.2.1 Stage 1: Signal Processing  
As we have seen, deep learning has advanced considerably in intelligent condition diagnosis, but it fails to preserve temporal dependency entirely when processing time domain vibration signals, which leads to a loss of data signals. Furthermore, existingnetworks such as recurrent neural network are difficult to train; hence, it is hard to construct accurate monitoring models. However, by converting time domain vibration signals into 2D image dataset through GAF, not only can enrich signal characteristics by filling vibration signals with pixels, but also establish bijective mapping between 1D vibration signals and 2D space to ensure the integrity of information.  
Since the inner product operation cannot retain both observed values during conversion process, both of them can only be converted into one value. To avoid this loss of information, Gram-like matrix is generated by calculating \(x\oplus y=\cos(\theta_{1}+\theta_{2})\), so that original values of scaled time series form diagonal lines, and 1D vibration signals are approximately reconstructed with advanced features extracted by deep learning, so as to maintain an absolute temporal relationship [50].  
Therefore, it is crucial to convert 1D time series signals into 2D images efficiently [51]. And the GAF is offered to convert 1D time series into RGB images in this study. Assume a time series \(X=\{x_{1},x_{2},\cdots,x_{n}\}\) of \(n\) real-valued observations, then rescale the time series \(X\) to keep all values are in the interval (-1, 1) or (0, 1) by:  
\[\tilde{x}_{-1}^{i}=\frac{(x_{i}-\max(X))+(x_{i}-\min(X))}{\max(X)-\min(X)} \tag{6}\]  
Figure 3: The flow diagram of the proposed approach.  
\[\tilde{x}_{0}^{i}=\frac{x_{i}-\min(X)}{\max(X)-\min(X)} \tag{7}\]  
where \(\tilde{x}_{-1}^{i}\) and \(\tilde{x}_{0}^{i}\) represent the values of \(x_{n}\) falling within the interval (-1, 1) or (0, 1).  
Therefore, the rescaled time series \(\widetilde{X}\) in polar coordinates can be obtained by encoding the value as the angular cosine and the time stamp as the radius, and the corresponding formula is as follows:  
\[\left\{\begin{array}{l}\varnothing=\arccos(\tilde{x}_{i}),-1\leq\widetilde{x }_{i}\leq 1,\widetilde{x}_{i}\in\widetilde{X}\\ r=\frac{t_{i}}{N},t_{i}\in\mathbb{N}\end{array}\right. \tag{8}\]  
where \(t_{i}\) is the time stamp and \(N\) denotes a constant factor to regularize the span of the polar coordinate system.  
The encoding map of Equation (8) has several advantages. First, since it is bijective, the original time series can be reconstructed by using \(\theta\) and \(r\). Second, the time dependence in the original time series is preserved through the coordinates of \(r\).  
Additionally, it can be seen from Equation (8) that (-1, 1) represents the cosine function in (0, \(\pi\)) and the cosine value decreases monotonically within this range. Likewise, (0, 1) denote the cosine function in \(\left[0,\frac{\pi}{2}\right]\). By calculating the cosine of the angle sum between various points, the GAF can generate two different images, i.e., the Gramian angular summation field (GASF) and the Gramian angular difference field (GADF). The two are, respectively, represented by the following formulas:  
\[GASF=\left[\cos(\phi_{i}+\phi_{j})\right]=\widetilde{X}^{\prime}\cdot \widetilde{X}-\sqrt{I-\widetilde{X}^{2}}\cdot\sqrt{I-\widetilde{X}^{2}} \tag{9}\]  
\[GADF=\left[\sin(\phi_{i}-\phi_{j})\right]=\sqrt{I-\widetilde{X}^{2}}\cdot \widetilde{X}-\widetilde{X}^{\prime}\cdot\sqrt{I-\widetilde{X}^{2}} \tag{10}\]  
where \(I\) denotes the unit row vector (1, 1, \(\cdot\cdot\cdot\), 1).  
In short, we only need to rescale the time series X into the polar coordinate system using Equation (8), then utilize the corresponding equation to calculate; finally, we can obtain the image from the GASF and GADF. The diagram of the above transformation is presented in Figure 4. In this study, we use GASF to convert 1D vibration signals into images.