# Extension of DBSCAN in Online Clustering: An Approach Based on Three-Layer Granular Models
## 5 Experiment and Discussion
### Synthetic Data
#### 5.1.2 Construction on Interval Granules in Medium Space
On the other hand, if the statistical distributions of _Error_is_ and _Error_not_ in Figures 8 and 9 are assumed to fit some probability density functions (PDF), then the objective functions in Equations (12) and (13) for calculating bounds are rewritten by the following continuous forms:  
\[\left\{ \begin{array}{l}f_{1}(\varepsilon)=\int\limits_{\pi\to\varepsilon}p( x)dx\\ f_{2}(\varepsilon)=\exp(-\alpha\cdot\varepsilon)\end{array} \right. \tag{21}\]  
Based on Equations (12), (13), and (21), the bounds of interval granules (_Error_is_ and _Error_not_) are calculated. According to the above description, the interval granule (_Error_is_), representing points within the \(i\)th cluster, is expressed as \(B_{i1}\), and the interval granule (_Error_not_) is defined as \(B_{i0}\) on the contrary. Their bounds in the medium space can be further expressed in detail, as (22):  
\[B_{i1}=[0,V(\varepsilon_{1}^{+})];\quad B_{i0}=[V(\varepsilon_{0}^{-}),V( \varepsilon_{0}^{+})]; \tag{22}\]  
where \(V(\varepsilon_{1}^{+}),V(\varepsilon_{0}^{-})\), \(V(\varepsilon_{0}^{+})\) are the optimal bounds calculated via GrC. The final numerical results are presented in Table 2, where IG\({}_{\text{y}}\) and IG\({}_{\text{n}}\) present the granular intervals of _Error_is_ and _Error_not_. Second, based on these constructed interval granules, we could build two fuzzy rules for each cluster, e.g., one rule for determining a data point belonging to a cluster, and one rule for not-belonging, as presented below:  
\[ \begin{array}{l}Rules\ for\ the\ i_{th}\ cluster\ C_{i}:\\ \left\{\begin{array}{l}R_{i}:\ IF\ Rec(\mathbf{x}_{k})\ is\ in\ B_{i1},\ THEN\ \mathbf{x}_{k}\ belongs\ to\ C_{i};\\ \overline{R}_{i}:\ IF\ Rec(\mathbf{x}_{k})\ is\ in\ B_{i0},\ THEN\ \mathbf{x}_{k}\ doesn^{\prime}t\ belong\ to\ C_{i};\end{array}\right.\end{array} \]  
where \(R_{i}\) is a rule used to determine if \(\mathbf{x}_{k}\) belongs to cluster \(C_{i}\); \(\overline{R}_{i}\) is the contrary rule used to determine that \(\mathbf{x}_{k}\) does not belong to cluster \(C_{i}\). These two rules are both fuzzy rules. With the help of these rules, when new data are obtained in online testing, clustering on these data can be quickly completed with less cost. Moreover, due to the fuzzy characteristics, the final decision can allow an extent of uncertainty when facing some confused data requiring manual checking, instead of making a determined decision blindly. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(c\) = **5** & \(c\) = **10** \\ \hline \multirow{2}{*}{Cluster1} & IG\({}_{\text{y}}\): [0, 0.0260]; & IG\({}_{\text{y}}\): [0, 0.0090]; \\  & IG\({}_{\text{n}}\): [0.0011, 0.2591]; & IG\({}_{\text{n}}\): [0.0034, 0.1914]; \\ \hline \multirow{2}{*}{Cluster2} & IG\({}_{\text{y}}\): [0, 0.0270]; & IG\({}_{\text{y}}\): [0, 0.0080]; \\  & IG\({}_{\text{n}}\): [0.0046, 0.2526]; & IG\({}_{\text{n}}\): [0.0055,0.1875]; \\ \hline \multirow{2}{*}{Cluster1} & \(c\) =15 & \(c\) =20 \\ \hline \multirow{2}{*}{Cluster2} & IG\({}_{\text{y}}\): [0, 0.0090]; & IG\({}_{\text{y}}\): [0, 0.0070]; \\  & IG\({}_{\text{n}}\): [0.0044, 0.1804]; & IG\({}_{\text{n}}\): [0.0036,0.1796]; \\ \hline \multirow{2}{*}{Cluster2} & IG\({}_{\text{y}}\): [0, 0.0070]; & IG\({}_{\text{y}}\): [0, 0.0070]; \\  & IG\({}_{\text{n}}\): [0.0047, 0.1887]; & IG\({}_{\text{n}}\): [0.0029, 0.1789]; \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of granular intervals in the medium space.