# Power Transformer Fault Diagnosis Based on Dissolved Gas Analysis by Correlation Coefficient-DBSCAN
## 2 DBSCAN Method  
DBSCAN algorithm was proposed by [PERSON], [PERSON] and others in 1996 [26]. It is a spatial clustering algorithm based on density. In order to accurately describe the algorithm, the following definitions are given first.  
Definition 1 given dataset \(D\), the _Eps_ neighborhood of an object \(p\) refers to the area with object \(p\) as the center and _Eps_ as the radius, that is  
\[N_{Eps}(p)=\{q\in D\big{|}Dist(p,q)\leq Eps\} \tag{1}\]  
where \(D\) is the dataset, _Dist(\(p\),\(q\))_ is the distance between object \(p\) and \(q\), which means all objects in dataset \(D\) that are not more than _Eps_ away from the object \(p\).  
Definition 2 given dataset \(D\), the core object refers to the object \(p\)\(\in\)\(D\) whose _Eps_ neighborhood has more objects than the given value _MinPts_, _MinPts_ is neighborhood density threshold, and core object \(p\) satisfies Equation (2):  
\[\big{|}N_{Eps}(p)\big{|}\geq\textit{MinPts} \tag{2}\]  
Definition 3 given dataset \(D\), direct density reachability is that if object \(q\) is in the _Eps_ neighborhood of object \(p\) and object \(p\) is the core object, then object \(q\) is direct density reachable to object \(p\), that is, there is object \(q\in N_{Eps}(p)\) and object \(p\) satisfies Equation (2).  
Definition 4 given dataset \(D\), density reachability is that if there is an object chain \(p_{1},p_{2},\ldots,p_{n}\in D\), for \(p_{i}(0<i<n)\), \(p_{i+1}\) is directly density reachable to object \(p_{i}\), then the object \(p_{n}\) is said to be density reachable to object \(p_{1}\).  
Definition 5 given dataset \(D\), density connection is that if there is object \(o\in D\), so that object \(p\) and object \(q\) are density reachable from object \(o\), then object \(p\) and object \(q\) are said to be density connected.  
Definition 6 given dataset \(D\), a cluster \(C\) is a non-empty subset of dataset \(D\), and the following conditions are met:  
1. For any object \(q\), if the core object \(p\)\(\in\)\(C\) and the object \(q\) is density reachable from the core object \(p\), then the object \(q\)\(\in\)\(C\).
2. For any object \(p\), \(q\)\(\in\)\(C\), object \(p\) and object \(q\) are density connected.  
Definition 7 given dataset \(D\), the noise point is the object that does not belong to any cluster.  
DBSCAN first performs region query on any object \(o\) to calculate its _Eps_ neighborhood. If  
\[\left|N_{\textit{Eps}}(o)\right|<\textit{MinPts}\]  
object \(o\) is temporarily marked as noise; otherwise, object \(o\) and its neighbors are marked as belonging to cluster \(C_{1}\), and region query is repeated for each neighbor of \(o\). When the former cluster \(C_{1}\) cannot be further expanded, any unmarked objects will be selected and a new cluster \(C_{2}\) will be generated based on the selected objects. Repeat for cluster \(C_{1}\) until all objects are marked.  
The essence of the density-based clustering algorithm is to find the high-density dataset in the dataset, that is, the average distance between data points in the dataset is small, while there are low-density areas between high-density datasets. The DBSCAN algorithm uses the parameters of _Eps_ and _MinPts_ to determine the threshold of dividing high-density datasets.