# A Method for Constructing a Synthesis Health Index for Metro Vehicle Wheelsets
## 1 Introduction
For certain devices or systems, a specific characteristic may dominate the degradation features; in other words, the degradation characteristics of the studied object can be represented by a single sensor parameter. In such cases, this dominant characteristic can serve as the PHI for that object. For example, the wheel diameter of a wheelset may be considered the PHI for assessing the wear status of the wheelset's diameter. However, relying solely on the PHI may overlook the influence of other characteristics on wheel diameter wear. In contrast, the SHI is constructed by integrating multiple PHIs or multi-sensor parameters, allowing for a more comprehensive assessment of the wheelset's health status. The SHI is primarily constructed through data fusion methods that integrate multi-dimensional signals into a single SHI, mathematically expressed as \(x(t)\overset{f(\cdot)}{\rightarrow}h(t)\), where \(x(t)\) represents the state data at time \(t\), \(h(t)\) is the SHI to be constructed at time \(t\), and \(f(\cdot)\) denotes the data fusion function. The primary methods for constructing the SHI include Principal Component Analysis (PCA), Self-Organizing Maps (SOMs), linear regression, statistical distance-based SHI construction methods, and neural networks. PCA, a widely used method for data dimensionality reduction, is often employed in constructing the SHI. [PERSON] et al. [5] developed an SHI for milling cutters based on PCA, considering both distance similarity and spatial direction similarity. [PERSON] et al. [6] utilized PCA to reduce feature dimensions and then calculated the deviation between the unknown state of bearings and their healthy state as the SHI. [PERSON] et al. [7] extracted three principal components from multi-dimensional data of bearings, including temperature, torque, and vibration, using PCA. They subsequently integrated these three components through Support Vector Data Description to track degradation states and detect early faults. Additionally, Kernel Principal Component Analysis (KPCA), an extension of PCA, has been employed to construct the SHI. [PERSON] et al. [8] automatically developed the SHI based on KPCA and adaptive fusion rules, validated through the 2008 PHM Challenge. SOMs are also widely utilized in the construction of the SHI. [PERSON] et al. [9] integrated multiple features of bearing vibration signals using SOMs, creating an SHI referred to as the \"Weighted Minimum Quantization Error\". Linear regression (LR) models are employed to fuse multi-dimensional data from failed systems and to convert monitoring data from in-service machines into the SHI. LR models are generally preferred over logistic regression for maintaining the system's initial degradation state [10]. [PERSON] et al. [11] trained the potential fault interval factor as the target SHI for the LR model. [PERSON] et al. [12] utilized the LR model to construct the SHI for aircraft engines. Typically, SHIs derived from regression contain noise, which can be reduced through smoothing and fitting operations. The key aspect of SHI construction methods based on statistical distance is the normalization of each dimension of the vectors. [PERSON] [13] constructed an SHI based on [PERSON]'s T\({}^{2}\) and performed RUL prediction for bearings. [PERSON] et al. [14] extracted statistical features from the wavelet coefficients of denoised bearing signals, employing Singular Value Decomposition and Kernel Fisher Discriminant Analysis for feature selection, ultimately constructing the SHI based on the Mahalaobis distance using the selected features. Neural networks, owing to their nonlinear fusion capabilities, are employed to construct the SHI. [PERSON] et al. [15] developed a time-series memory Auto Encoder (AE) that enhances SHI representation by sequentially updating reconstructions to capture temporal dependencies. [PERSON] et al. [16] proposed a supervised multi-head self-attention network that improved the monotonicity and robustness of the SHI, along with an unsupervised model for SHI extraction. [PERSON] et al.