## 2 Methodologies
### Model Fusion and Evaluation  
Model fusion merges multiple weak models into a strong model, which reduces the tendency of a single model to be overfitted. The fusion of multiple models can improve the normalization ability. It avoids the case wherein the single model is unstable with a low fault diagnosis ability. Multiple models can often improve the fault diagnosis ability and the robustness of the model significantly. Waterfall model fusion uses a method of connecting multiple models in series. Each recommendation algorithm is regarded as a model. Waterfall model fusion connects models with different functions back and forth, as shown in Figure 9.  
Figure 8: Working principle of the SDAE. In the waterfall fusion, the target result achieved by the previous recommendation method can be used as the input of the latter method, progressively step by step. The function of the fusion model is gradually enhanced, and finally a result set with a low quantity and high quality is obtained. It is usually used to implement the recommended scenarios for different model functions. In designing a waterfall model-fusion method, the model functions are usually sequenced and gradually transitioned to achieve the final goal. In the face of a large number of candidate recommendation objects, but few valuable recommendation results, high accuracy requirements and limited computing time, it is often suitable. The small fault-sample regeneration model (GAN) and SDAE model are a series of two single models, adopting the waterfall-type in model fusion. The GAN-SDAE structure is shown in Figure 10. The GAN-SDAE system based on waterfall model fusion strengthens the model's fault tolerance rate and improves the accuracy of the integrated learning. For the GAN-SDAE system, the number of neurons is more significant than the layer thickness in the case of small fault samples [38]. The number of layers for both GAN and SDAE is three or four each. To construct an optimal GAN-SDAE system structure, a reconstruction error in the deep networks is proposed [39]. The reconstruction errors of each sample from the GAN-SDAE were obtained in a training process for a fault diagnostic task. Because the real fault diagnosis result (C) for each sample is known, the diagnostic error can be calculated. The output result from the GAN-SDAE system is \(Z\), and the reconstruction error \(\varepsilon\) in the training process is given by \(\varepsilon=|C-Z|\). Different reconstruction errors for different network structures were calculated. The network structure having the least reconstruction error was finally selected as the optimal structure (the various  
Figure 10: Waterfall model fusion for the GAN-SDAE system. Figure 9: Waterfall model fusion. network structures of GAN-SDAE and corresponding reconstruction errors are analyzed in Section 3). Combining the analyses in Sections 2.2 and 2.3 provides an overall flow chart of using GAN-SDAE for machine fault diagnosis; this is shown in Figure 11.  
Evaluation indicators are regarded as developers showing the results to users and making standardized comparisons. Moreover, evaluation indicators can illustrate the appropriate expression of needs and scientific information. Over the years, due to the practice of fault diagnosis methods in various disciplines, various index sets have been established in the field of fault diagnosis to evaluate model performance [40]. The metric sets are mainly designed to verify the performance of the GAN-SDAE system's applications. Therefore, evaluation indicators are particularly important in the model development phase, which can be used to improve the fault diagnosis program. The error is basically expressed as a deviation from the desired target. Most of the accuracy-based metrics are derived directly or indirectly from the error term. The error is expressed as  
\[\text{Error}=e_{i}=y_{i}-\hat{y}_{i} \tag{13}\]  
where \(\hat{y}_{i}\) is the estimated value and \(y_{i}\) is the expected output value.