## 2 Materials and Methods
### EONND: Energy Operator Detector with Neural Network Architecture
#### 2.3.1 Convolutional Layer  
As shown in Figure 2, in the first step, the signal is converted into the form of a \(k\times k\) Toplitz matrix by a multi-resolution energy operator. The Toplitz matrix is determined  
Figure 1: Schematic diagram of the correspondence between the functional modules of CNN and traditional change-point detector algorithms.  
Figure 2: The architecture of EONND.  
by the elements of the first row and the first column. In this paper, the first row was \([x(n)\ x(n+1)\ x(n+\cdots)\ \ldots\ x(n+k)]\), and the first column was \([x(n)\ x(n-1)\ x(n-\cdots)\ \ldots\ x(n-k)]\), where \(x(n)\) represents the \(n\)th sampling point. Based on the sampling points calculation relationship established by Equation (3), the convolution layer of EONND is shown in Figure 3.  
Figure 3 illustrates the logic of EONND for sampling-points processing in the form of the most common neurons in neural network algorithms. This part of the algorithm has the function of signal filtering and enhancing the active segment features, since the computational relationship between the sampling points is a multi-resolution energy operator. More specifically, \(f(x)=x^{2}\) is used to replace ReLU as the activation function, and a local connection is used to establish the computational relationship for the data in the range \(k\) before and after sampling point \(n\). Compared with the traditional TKEO approach of observing only three sampling points to obtain the \"energy\" of the signal, Equation (7) allows the energy operator to observe more contextual content and thus obtain more accurate features of sampling point \(n\).  
\[C(k)=W_{1}(w_{11}x(n+k))^{2}+W_{2}(w_{22}x(n))^{2}+W_{3}(w_{33}x(n-k))^{2}+W_{4 }(w_{14}x(n+k)+w_{34}x(n-k))^{2} \tag{7}\]  
The weights parameter \(W_{1-4}\) in Equation (7) is updated by training with the traditional neural network algorithms, while in this paper, the parameter \(W\) was determined by establishing the relationship of Equation (8) with the multi-resolution energy operators.  
\[C(k)=\Psi_{k}[x(n)]=x^{2}(n)-x(n+k)x(n-k) \tag{8}\]  
Then, we were able to obtain: \(W_{1}=W_{2}=W_{3}=1\), \(W_{4}=-1\), \(w_{11}=w_{14}=2\), \(w_{22}=1\), and \(w_{33}=w_{34}=1/4\).  
In fact, after determining the neuron parameters by Equation (8), the time-lag value \(k\) becomes the only hyperparameter in CONV that affects the performance of EONND. According to previous studies, the multi-resolution energy operator with odd values in the  
Figure 3: Schematic diagram of CONV.  
range of the scale factor \(k=15\) can cover the signal \"energy\" up to 450 Hz [36; 39; 47; 48; 49]. This paper set \(k=15\), so that for each sampling point \(n\), we could obtain odd values within 15 [\(C(1)\), \(C(3)\), \(C(5)\),..., \(C(15)\)], for a total of 8 possible \"energy\" features.  
As shown in Figure 3, CONV processes \(2k+1\) sampling points at a time and outputs possible \"energy\" features as input to the pooling layer through a local connected layer and a global connected layer. In addition, all parameters within CONV were fixed by the relational Equation (8).