## 4 Evolutionary Characteristics of Major Fractures
### Method for Determining DBSCAN Parameters  
DBSCAN algorithm is a density-based clustering algorithm that does not require a pre-set number of clusters. However, it requires pre-determination of two parameters, \(\varepsilon\) and minPts, which describe the density of the data points. The clustering results are highly sensitive to the values of \(\varepsilon\) and minPts. Choosing inappropriate values may result in poor clustering performance or even clustering errors. One commonly used method to determine the values of \(\varepsilon\) is the \(k\)-nearest distance method, which involves the following steps:  
1. The distances between objects in the set \(D\) are calculated to obtain the distance matrix \(\mathit{Dist}_{n\times n}\) \[\mathit{Dist}_{n\times n}=\left\{\mathit{dist}(i,j),1\leq i\leq n,1\leq j\leq n\right\}\] (1) where \(n\) represents the number of data points in the uncategorized dataset \(D\), \(\mathit{Dist}_{n\times n}\) is a real symmetric matrix with \(n\) rows and \(n\) columns, \(\mathit{dist}(i,j)\) represents the distance from the \(i\)-th object to the \(j\)-th object in set \(D\), and \(\mathit{dist}(i,j)\) is calculated using the Euclidean distance, expressed as Formula (2). \[\mathit{dist}(i,j)=\sqrt{\sum_{k=1}^{m}\left(x_{ik}-x_{jk}\right)^{2}}\] (2)  
Figure 7: Illustration of the basic concept of DBSCAN.  
where \(m\) represents the number of features for each element, \(x_{ik}\) denotes the value of the \(k\)-th feature parameter of the \(i\)-th element, and \(x_{jk}\) represents the numerical value of the \(k\)-th feature parameter of the \(j\)-th element.
2. Each row of the matrix \(Dist_{n\times n}\) is sorted in an ascending order. Each row represents a ranking of the distances from the corresponding data point to all the other points. After sorting, \(Dist_{n\times k}\) will be a collection of the \(k\)-th closest distance values to each data point.
3. Column of the matrix \(Dist_{n\times n}\) is sorted in an ascending order, and the curve graph of the \(k\)-th column (referred to as the \(k\)-curve) is plotted. The distance corresponding to the steep inflection point of the \(k\)-curve is the \(\varepsilon\) determined using the \(k\)-nearest neighbor distance method. By analyzing the change in the steep inflection points as \(k\) increases, it is observed that after a certain threshold, steep inflection points of \(k\)-curve concentrate in a specific region; this indicates that with the increase in \(k\), outlier points remains essentially the same, and both the clustering point and outlier point detection results tend to stabilize ([PERSON] et al. (2020)). The distance value corresponding to the region of concentration of steep inflection points in the \(k\)-curve is taken as the \(\varepsilon\) value.
4. The expectation method is used to generate MinPts. For \(\varepsilon\), the number of objects within the \(\varepsilon\)-neighborhood of each data point is calculated sequentially. The mathematical expectation of the number within \(\varepsilon\)-neighborhoods for all data points is then computed, and this value is taken as the MinPts for set \(D\).  
\[MinPts=\frac{1}{n}{\sum_{i=1}^{n}P_{i}} \tag{3}\]  
where \(P_{i}\) represents the number of objects within the \(\varepsilon\)-neighborhood of the \(i\)-th data point, and \(n\) represents the total number of data points.