# Probability Analysis of Hypertension-Related Symptoms Based on XGBoost and Clustering Algorithm
## Materials and Methods
### Clustering Methodology
Gaussian kernel:  
\[\rho_{i}=\sum_{j\in I_{k}\backslash\{i\}}e^{-(\frac{d_{ij}}{d_{c}})^{2}} \tag{2}\]  
Compared with the Formula (1), the cut-off kernel is a discrete value, but the Gaussian kernel is a continuous value, so the probability that different data points have the same local density is smaller under the latter condition, and the latter also satisfies the property that the closer to the data point, the greater the density. Distance \(\delta_{i}\):  
Assuming that \(\{q_{i}\}_{i=1}^{N}\) represents a descending order of \(\{\rho_{i}\}_{i=1}^{N}\), it means that it satisfies:  
\[\rho_{q1}\geq\rho_{q2}\geq\cdots\geq\rho_{qN}\]  
Defined as follows:  
\[\delta_{q_{i}}= \begin{cases}\underset{q_{i}}{\min}\{d_{q_{i}q_{j}}\},&i\geq 2 ;\\ \underset{j<i}{\max}\{\delta_{q_{i}}\},&i=1.\end{cases}  \tag{3}\]  
According to the Formula (3), when the local density of \(x_{i}\) is the largest. \(\delta_{i}\) represents the distance of the point in \(S\) where the distance from \(x_{i}\) is the largest, otherwise \(\delta_{i}\) represents the distance from the point where the distance from \(x_{i}\) is the smallest among the points where the local density is larger than \(x_{i}\). So far, for each data point \(x_{i}\) in \(S\), we can calculate \((\rho_{i},\delta_{i}),i\in I_{s}\). For example, in a two-dimensional space, 20 points are distributed, and the numbers in the circles represent the number of points, and the distribution of points is shown in Figure 1A. The \(\rho\) and \(\delta\) of each point are calculated and represented in a two-dimensional space, as shown in Figure 1B. \(\rho\) is the horizontal axis and \(\delta\) is the longitudinal axis. Figure 1B plays a decisive role in the selection of clustering centers, because subjective selection plays a great role in the selection of clustering centers, it is necessary to create a new metric \(\gamma\), the product of \(\rho\) and \(\delta\), the greater the product, the higher the probability of being selected as a clustering center. \[\gamma_{i}=\rho_{i}\delta_{i},iels \tag{4}\]  
The algorithm flow chart of this method is as follows. Step 1: Initialization and Preprocessing  
* Given a parameter \(t\in(0,1)\) which is the same as determining the cut-off distance \(d_{c}\)
* Calculate the distance \(d_{ij}\), and make \(d_{ij}=d_{ji},i<j,i,j\in I_{s}\)
* Determining cut-off distance \(d_{c}\)
* Compute \(\{\rho_{i}\}_{i=1}^{N}\) and generate its descending order subscript \(\{q_{i}\}_{i=1}^{N}\)
* Calculate \(\{\delta_{i}\}_{i=1}^{N}\) and \(\{n_{i}\}_{i=1}^{N}\)  
Step 2: Select the cluster center \(\left\{m_{j}\right\}_{j=1}^{n_{c}}\) and initialize the data point classification attribute tag \(\left\{c_{j}\right\}_{i=1}^{n}\), as follows:  
\[c_{i}=\left\{ \begin{array}{ll}k,&\text{If x}_{i}\text{ is the distance center, it belongs to the first k cluster.}\\ -1,&\text{\emph{other}}\end{array} \right.\]  
Step 3: Categorization of non-clustered central data points  
Step 4: If \(n_{c}>1\), the data points in each cluster are further divided into cluster core and cluster halo * Initialization mark \(h_{i}=0,i\in I_{S}\)
* Generate an average local density upper bound \(\left\{\rho_{i}^{\mathrm{p}}\right\}_{i=1}^{n_{c}}\) for each cluster. * Identification cluster halo