## 2 Brief Explanation of Proposed Diagnosis Method
### Construction of CNN for Fault Diagnosis  
Typically, a CNN is composed of a feature extraction layer and a feature mapping layer. Neurons play a significant role in connecting these layers. The feature extraction layer consists of a local acceptable domain of the previous layer, which is connected through neurons. The local characteristics are extracted and forwarded to the feature mapping layer, which consists of multiple feature maps. Generally, a feature map is a plane filled with neurons, and it is followed by a calculation layer, which is used to calculate the features. The basic structure of a CNN model consists of input, hidden, and output layers. They are fully interconnected in one direction, from the input layer to the output layer, including a hidden layer and a pooling layer.  
In our present study, the general structure of a CNN model was modified by adding an auto-tuning function. This auto-tuning function made adjustments to the hidden layer and pooling layer considering the accuracy rate and overlapping percentage. The details of the auto-tuning function are explained in the Procedures and Results of Diagnosis section. In our study, forward propagation in the CNN was performed, and the neurons in the hidden layer were decided based on the input and output patterns that must be recognized. The neurons in the hidden layer performed mathematical operations considering the number of inputs, and the post-mathematical operation results were given to output layer, where the final prediction was performed. These neurons of the hidden layer were expressed as the weight matrix and connected the adjacent layers (input and output layers). The output of the hidden layer can be mathematically expressed using Equation (1):  
\[Z=W^{T}X+b \tag{1}\]  
where \(W\) is weight expressed in matrix form, \(T\) represents transposition, \(X\) denotes input to the hidden layer, and \(b\) is the bias constant.  
Apart from the basic model, in the proposed diagnosis system, an encoder was added with an auto-tuning function. In the basic model of a CNN, both the auto-tuning function and encoder are not attached; as a result, for each overlapping condition, a low accuracy rate is obtained. With the addition of auto-tuning and an encoder, automatic optimization of the hidden layer and pooling layer was made possible, and the missing data count was reduced. The results, evaluation, and drawbacks of the proposed method are discussed in Section 4.  
The structure of the model depends on the number of input data. In the present report, three types of bearing conditions are discussed (healthy, SS, and FS) and the structure of the encoder is framed considering three input data. The auto-encoder consisted of two parts: an encoder and a decoder. The encoder was used to extract features from the input data, which were given to the decoder for reconstructing the sampling data from the hidden layer. The weight parameters and the encoder and decoder bias vectors were adjusted according to the number of inputs, and the outputs were predicted.  
A three-layer feed-forward propagation CNN was selected; both the input and output were defined as multiple variables without clear linear relationships. Considering the faulttypes and the number of dependencies, the proposed model consisted of three pooling layers and three hidden layers. A convolution layer was used for modeling and classification. A pooling layer was added to the basic structure to reduce the overlapping effects and to perform filtering and simple discretization. The proposed CNN model for bearing failure diagnosis is shown in Figure 1.