## 3 Preliminaries
### Optics  
OPTICS is a density-based clustering algorithm, which is improved on DBSCAN. Because the DBSCAN algorithm is very sensitive to the initial parameters, the influence of the first two parameters on the calculation results is crucial [41]. Some concepts of the OPTICS algorithm are defined in the same way as DBSCAN, such as \(\sigma\)-Neighborhood, core object, direct density, density reachability, and density connection [42] (assuming our sample set is):  
\(\sigma\)-Neighborhood: For \(x_{i}\in\ X\), its \(\sigma\)-neighborhood contains the sub-sample set whose distance between \(x_{j}\) and in the sample set is not greater than \(\sigma\). \(\sigma\)-Neighborhood is a set, expressed as follows, the cardinality of this set is recorded as \(\big{|}N_{\sigma}\big{(}x_{j}\big{)}\big{|}\).  
\[N_{\sigma}\big{(}x_{j}\big{)}=\big{\{}x_{i}\in X\ |\ \ \text{distance}\ \big{(}x_{i},x_{j}\big{)}\leq\sigma\big{\}} \tag{3}\]  
Core object: For any sample \(x_{i}\in X\), if its \(\sigma\)-Neighborhood corresponds to \(N_{\sigma}\big{(}x_{j}\big{)}\) at least contains MinPts samples, if \(\big{|}N_{\sigma}\big{(}x_{j}\big{)}\big{|}\geq MinPts\), then \(x_{j}\) is the core object.  
Density direct: If \(x_{i}\) is located in the \(\sigma\)-neighborhood of \(x_{j}\) and \(x_{j}\) is the core object, it is said that \(x_{i}\) is to be direct by density \(x_{j}\). The opposite is not necessarily true; it cannot be said to be directly reached by density at this time unless \(x_{i}\) is also the core object, and direct density does not satisfy symmetry.  
Density is reachable: for \(x_{i}\) and \(x_{j}\), if there is a sample sequence \(p_{1}\), \(p_{2}\),..., \(p_{r}\) that satisfies \(p_{1}=x_{i}\), \(p_{r}=x_{j}\) and \(p_{t+1}  
Reachability-distance: Let \(x,y\in X\), for a given \(\sigma\) and \(MinPts\), the reachability-distance of \(y\) about \(x\) is defined as:  
\[rd(y,x)=\left\{\begin{array}{c}undefined|N_{\sigma}(x)|<MinPts\\ max\{cd(x),d(x,y)\}|N_{\sigma}(x)|\geq MinPts\end{array}\right. \tag{5}\]  
In particular, when \(x\) is the core point (the corresponding parameters), it can be understood \(rd(y,x)\) according to the following Formula (6):  
\[rd(y,x)=min\{\eta:y\in N_{n}(x)\&\big{|}N_{\eta}(x)\big{|}\geq MinPts\} \tag{6}\]  
The Birch algorithm is a traditional and efficient hierarchical clustering algorithm that does not require pre-setting the entire dataset, due to the fact that the Birch algorithm scans the dataset in an incremental and dynamic manner to construct the clustering feature tree (CF tree) of the dendrogram [43]. Hierarchical coalescence and iterative relocation are combined by it, using a bottom-up hierarchical algorithm, and then the results are updated using iterative relocation.  
It has two main stages, including the construction of a memory tree by scanning the dataset and then applying the algorithm to the cluster leaf nodes. The CF tree is a height balanced tree controlled by two parameters, a branching factor B and a threshold T, constructed while scanning the data. When a data point is encountered, the CF tree is traversed by selecting the nearest node at each level starting from the root.