## 2 Methodology
### Vmd-CsANN Joint Model Construction  
The \(N\) segments of raw acceleration signal time series obtained from different measurement points by sensor devices can be represented as an \(N*T\) dimensional matrix \(M\). The length of this time series is determined by the sampling time and frequency. Each segment of the time series \(S_{i}\) is variational mode decomposition into \(K\) intrinsic mode function (IMF) components \(v_{k}(t)\) using the measurement point as a reference. Based on the decomposed central frequency \(\omega_{t}\), the IMF components are recombined to form new local signal sequences \(S_{t}\). These new signal sequences are concatenated with the original signal sequences, using time as a reference, to create feature signals \(S_{i,t}\). The combination of all feature signal sequences forms a new feature matrix \(M_{t}\).  
Figure 2: Basic calculation process of the convolutional layer.  
After each vector matrix data is preprocessed as input data, it is input into CSANN. The convolution layer extracts the signal features of the data through the convolution operation of the input data, as shown in Figure 4.  
The convolution layer begins by initializing \(N\) 1-dimensional convolution kernels of length \(x\), employing Gaussian functions for random mobilization. These convolution kernels then slide horizontally along the input one-dimensional data matrix with a defined sliding interval. At each position, a product operation and summation is performed on the element values, resulting in a value. Ultimately, the m \(\times\) 1-dimensional one-dimensional data matrix is transformed into a \([(\rm{m-x})/k+1]\times\rm{n}\)-dimensional output matrix, denoted as \(S\).  
Figure 4: VMD-CSANN model construction.  
Figure 3: Schematic diagram of the self-attention mechanism layer.  
During the linear operation of data replication, it is customary to perform activation calculations on the output matrix \(S\). This entails coupling each element with an active element in the matrix \(S\), thereby introducing nonlinear factors. The resulting matrix is then transferred to the next convolution layer for further processing, as follows:  
\[f(i)=s(i)\times K \tag{9}\]  
In Equation (9), \(f(i)\) represents each data value in the output matrix \(S\) after the addition of factors, \(s(i)\) represents each data value in matrix \(S\), and \(K\) represents the activation function utilized by the convolutional layer.  
In the construction of the convolutional neural network model, it is customary to insert pooling layers between adjacent convolutional layers to mitigate overfitting. These pooling layers typically include MaxPooling and AveragePooling. For this study, we employed one-dimensional maximum pooling to process the time series data from the convolution operation, denoted as time series \(A\). \(A\) was segmented into several informational fragments \(A_{i}\), from which the maximum element value was extracted to form a new input matrix, denoted as \(B\):  
\[a_{i,max}=\max\{a_{1},a_{2},\cdots,a_{i}\} \tag{10}\]  
In Formula (10), \(a_{i,max}\) are the data obtained after maximum pooling of \(a_{i}\) from the original sequence \(A\).