## 5 Rolling Bearing Reliability Model
### PaCMAP  
Pairwise Controlled Manifold Approximation Projection (PaCMAP) is an efficient and robust nonlinear dimensionality reduction algorithm designed to embed high-dimensional data into low-dimensional space while preserving the geometric and topological characteristics of the data to the greatest extent. At present, it is used by many scholars in the field of data dimensionality reduction [32; 33]. By optimizing pairwise distance relationships between data points, the algorithm balances the preservation of local neighborhood structures and global distribution patterns, effectively maintaining data integrity and revealing the intrinsic features. Compared to other dimensionality reduction methods, PaCMAP demonstrates superior computational speed and the ability to retain global structures, making it particularly  
Figure 2: IVY-CYCBD flow chart. suitable for reducing the dimensionality of high-dimensional complex data. The algorithm follows three main steps:  
1. Construction phase: PaCMAP uses edges to construct the basic structure of a graph. PaCMAP distinguishes three types of edges: neighbor pairs, medium-near pairs, and distant pairs. The first type of edge includes the \(n_{\text{NB}}\) nearest neighbors of each sample point in high-dimensional space. The following scaled distance metric is used for calculation: \[d_{ij}^{2,\text{select}}=\frac{\left\|\mathbf{x}_{i}-\mathbf{x}_{j}\right\|^{2 }}{\sigma_{ij}}\quad,\sigma_{ij}=\sigma_{i}\sigma_{j}\] (19) where \(\sigma_{i}\) represents the average Euclidean distance from sample point \(i\) to its fourth to sixth nearest neighbors. This scaled distance is used to construct neighbor pairs \((i,j_{t}),t=1,2,\ldots,n_{NB}\). The scaling process is designed to address the issue of significant differences in neighborhood scales that may occur in different regions of the feature space. Therefore, the scaled distance \(d_{ij}^{2,\text{select}}\) is used only for the neighbor selection process and not for subsequent optimization stages. The second type of edge includes \(N\cdot n_{MN}\) medium-near pairs randomly selected from each sample, with an additional six observations randomly drawn from each observation value, and the second smallest medium-near pair among them is used. The third type of edge consists of \(n_{FP}\) distant pairs randomly selected from the sample each time. The number of medium-near pairs and distant pairs is determined by the parameters \(MN_{\text{ratio}}\) and \(FP_{\text{ratio}}\), which specify the ratio of the number of these pairs to the number of nearest neighbors, that is, \(n_{MN}=MN_{\text{ratio}}\cdot n_{NB}\) and \(n_{FP}=FP_{\text{ratio}}\cdot n_{NB}\). PaCMAP uses three different loss functions for each type of pair: \[Loss_{NB}=\frac{\tilde{d}_{ij}}{10+\tilde{d}_{ij}}\] (20) \[Loss_{MN}=\frac{\tilde{d}_{k}}{10,000+\tilde{d}_{k}}\] (21) \[Loss_{FP}=\frac{1}{1+\tilde{d}_{il}}\] (22) where \(\tilde{d}_{cd}=\|y_{c}-y_{d}\|^{2}+1\). The loss function for these edge pairs is determined by the overall loss weighted by \(w_{NB}\), \(w_{MN}\), and \(w_{FP}\). 2. Initialization: Although the initialization method has little impact on the results of PaCMAP, Principal Component Analysis (PCA) is still used to reduce actual running time. 3. Iterative Optimization: The iterative process is divided into three phases to avoid falling into local optima. In the first phase (from the first iteration to the \(N\)th iteration), the weight of the medium-near pairs is gradually reduced with the goal of improving the initial positions of the embedding points, preserving both global and local structures to some extent. In the second phase (from the \(N\)th iteration to the \(M\)th iteration), the objective is to refine the local structure by assigning a small (non-zero) weight to the medium-near pairs, while maintaining the global structure captured in the first phase.