## 2 Related Work
### Techniques for Converting Time Series Data into 2D Representations  
#### 2.3.1 Methods Based on Data Rearrangement  
A time series consists of data points that change over time, with each point corresponding to an observation or measurement taken at a specific moment. Unlike images, time series data are predominantly one-dimensional and demonstrate continuity characteristics that evolve over time. Typically, time series data comprise a series of observations arranged chronologically, revealing temporal correlations and trends among the data points. In industrial settings, vibration signals generated by machinery during operation are a representative form of time series data. In coal mining operations, bearing vibration signals are a common source of raw data for diagnosing machinery faults. These vibration signals are sampled continuously over time, capturing variations in amplitude and frequency generated by the bearings during operation. By employing deep learning models to analyze these vibration signals and automatically extract fault-related features, various potential fault patterns in bearings can be identified, including inner ring wear, outer ring cracks, and rolling element detachments ([PERSON] et al., 2019; [PERSON] et al., 2020). As a type of time series data, the original vibration signal is typically composed of data points arranged in chronological order. To optimize the use of deep learning methods for analyzing vibration signals, data rearrangement techniques can be employed to convert the signals into a two-dimensional matrix format. These methods generally involve first segmenting the data samples by defining a fixed time window or applying a specific strategy for overlapping sampling. Subsequently, the time series data are rearranged into a two-dimensional matrix according to predetermined transformation rules, allowing each sample to be represented as a two-dimensional image. This approach enhances compatibility with deep learning models, such as convolutional neural networks. Prior to data rearrangement, amplitude normalization is often applied as a preprocessing step to improve the comparability and stability of vibration signal data. This process involves scaling the amplitude range of the original vibration signals to a fixed interval, typically within \([-1,1]\) or \([0,1]\). By normalizing the amplitude, discrepancies between different vibration signals can be minimized, facilitating comparison and analysis within the same amplitude range. This step also helps mitigate issues of gradient explosion or vanishing gradients during the neural network training process. For a single-source raw vibration signal, the rules for rearrangement vary depending on the sampling method employed. There are generally two sampling approaches: one involves segmenting the vibration signal into several fixed-length segments along the time axis using a fixed-length time window, while the other employs overlapping sampling, in which there is a certain degree of overlap between adjacent time windows. The sampling processes for both methods are illustrated in Figure 1. After sampling is completed, the vibration signal data within each time window are sequentially arranged in the rows or columns of the matrix, resulting in a two-dimensional matrix. Specifically, let \(X\) represent a raw vibration signal with a length of \(T\). When employing a fixed-size time window \(L\) for data segmentation, the original data can be divided into \(N\) time segments. The segments resulting from this division are denoted as \(S\), such that  
\[S_{ij}=X_{j+(i-1)\times L} \tag{1}\]  
where \(i\) represents the \(i\)-th segment of the signal \(X\) for \(i=1,2,\cdots,N\), and \(j\) denotes the \(j\)-th data point. As to overlapping sampling, let \(K\) represent the length of the overlap and denote the segments obtained from this division as \(S^{o}\). Then,  
\[S^{o}_{ij}=X_{j+(i-1)\times K} \tag{2}\]  
Assuming that the vibration signal within the time window is arranged by rows and columns.