## 3 Random Convolution Kernel  
As representative algorithms in deep learning, CNNs have a wide range of applications and can be used to process data in different dimensions, including one-dimensional time series, two-dimensional images, and three-dimensional videos. One-dimensional CNNs have proven to be highly effective in processing time series data acquired from sensors. A CNN typically uses two primary components: feature extraction and pattern classification. In the feature extraction stage, convolutional and pooling layers are commonly used, often accompanied by an activation layer to enhance the extraction of key features. Subsequently, a fully connected layer is employed to perform the pattern classification task. This architectural approach enables CNNs to excel in many data-processing tasks, making them versatile tools for deep learning. The convolutional kernel serves as the central component of the convolution layer, performing a sliding convolution operation on the input data with a step size of 1, as shown in Figure 5. This allows for weight sharing, feature extraction, and improved computational speeds. The formula for one-dimensional convolution is as follows [13]:  
\[f\ =\ \mathbf{w}\mathbf{x}+b \tag{5}\]  
Figure 4: DTW algorithm divides the data flow diagram. where \(\mathbf{w}\) is the weight matrix of the convolution kernel, \(\mathbf{x}\) represents the input data, and \(b\) represents the bias. CNNs employ convolutional kernels to efficiently capture diverse features and patterns within time series data through convolutional operations. The use of a large number of random convolutional kernels enhances the ability of the network to identify discriminative patterns within a time series. The essential parameters of a convolutional kernel include the length, weight, bias, kernel dilation, and padding. A substantial number of random convolution kernels were utilized to achieve an effective transformation of the time series, each configured with specific parameter values. The lengths of the convolutional kernels were randomly selected with equal probability from among \(\{7,9,11\}\), and the lengths used were typically much shorter than those of the input time series. The weights \(w_{i}\) of the convolutional kernels were randomly drawn from a standard normal distribution. These weights are generally modest in magnitude but can potentially assume larger values. \[\forall w_{i}\;\epsilon\;\mathbf{w},\;w_{i}\sim N(0,\;1) \tag{6}\]  
where \(\mathbf{w}\) is the weight matrix of the convolution kernel. The bias term \(b\) is determined through random sampling from a uniform distribution, \(b\;\sim U(-1,\;1)\). Notably, distinct bias values were assigned, even when dealing with convolutional kernels that were otherwise similar. This biased divergence contributed to the extraction of diverse features from the input data. The kernel dilation parameter is pivotal to effectively enable a convolution kernel to capture patterns or features across diverse scales. The dilation rate \(d\) for kernel dilation was determined by random sampling, which typically follows the following distribution:  
\[d\;=\;\lfloor 2^{x}\rfloor,\;x\sim U(0,\;A),A\;=\;\log_{2}\frac{l_{input}-1}{ l_{kernel}-1} \tag{7}\]  
where \(l_{input}\) is the length of the input data and \(l_{kernel}\) is the length of the convolution kernel. This random sampling of the dilation rate ensures that the convolution kernel can accommodate patterns or features with varying frequencies and scales. Furthermore, during the generation of each convolutional kernel, a random decision (with equal probability) is made to determine whether a padding operation should be performed. If padding is selected, a certain amount of zero padding is added at the beginning and end of the input time series when the convolution kernel is applied. This ensures that the \"center\" element of the convolution kernel aligns with every point in the time series.