## 5 Proposed Model
### Density-Based Clustering
#### 5.2.1 Division Stage  
At the first subdivision stage, the WSN nodes are split up into several clusters using a k-means algorithm with dissimilar k-values. For instance, consider WSN nodes with r-observations \(\{\text{WSN}_{1},\ \text{WSN}_{2},\ldots,\text{WSN}_{\text{r}}\}\) in which every observation had g-dimensions of \(\text{WSN}_{\text{j}}=\left\{\text{WSN}_{\text{j}}^{\text{i}},\text{WSN}_{ \text{j}}^{\text{j}},\ldots,\text{WSN}_{\text{j}}^{\text{g}}\right\}\). The split dimension data can be further split using the k-means algorithm, which can be formulated as:  
\[\text{WSN}^{\text{i}\text{k-means}}=\left\{\text{clu}_{\text{i}}^{\text{i}}, \text{clu}_{\text{2}}^{\text{i}},\ldots,\text{clu}_{\text{k}}^{\text{i}} \right\};\text{k}=1,2,\ldots,\text{t} \tag{6}\]  
For every k-cluster and t \(\geq\) 1, the cluster variance can be computed by summing the split clusters, which can be formulated as:  
\[\mathbb{V}_{\text{clu}^{\text{i}}}(\text{k})=\sum_{\text{j}=1}^{\text{k}} \text{var}\Big{(}\text{clu}_{\text{j}}^{\text{i}}\Big{)};\text{k}=1,2,.., \text{t} \tag{7}\]  
In Equation (7), the normalized variance can be computed for \(\mathbb{V}_{\text{clu}^{\text{i}}}=\{\mathbb{V}_{\text{clu}^{\text{i}}}(1), \mathbb{V}_{\text{clu}^{\text{i}}}(2),\ \ldots,\mathbb{V}_{\text{clu}^{\text{i}}}(\text{t})\}\) and can be formulated as:  
\[\mathcal{D}_{\text{clu}^{\text{i}}}(\text{k})=\left\{\frac{\left|\mathbb{V}_{ \text{clu}^{\text{i}}}(\text{k}+1)-\mathbb{V}_{\text{clu}^{\text{i}}}(\text{ k})\right|}{\mathbb{V}_{\text{clu}^{\text{i}}}(\text{k})}:\text{k}=1,2,..,\text{t}-1\right\} \tag{8}\]  
The finest clusters are obtained from the \(\mathcal{D}_{\text{clu}^{\text{i}}}\) for every dimension that can be formulated as:  
\[\text{fin}^{\text{i}}=\underset{\text{i}}{\text{argmax}}\Big{\{}\text{k}: \mathcal{D}_{\text{clu}^{\text{i}}}(\text{k})\geq\text{Th}^{1}\Big{\}} \tag{9}\]  
In Equation (9), the piercing threshold value can be denoted as Th\({}^{1}\). If the \(\mathcal{D}_{\text{clu}^{\text{i}}}\) is equal or greater than the Th\({}^{\text{ini}}\), the corresponding dimension is taken as fin\({}^{\text{i}}\). After that, the fin\({}^{\text{i}}\) are further subdivided into several sub-clusters using the k-means algorithm.  
Followed by the first subdivision stage, the second subdivision stage is initiated. In that stage, the sub-clusters obtained from the first subdivision stage are divided based on the steps followed in the first subdivision stage. To be clearer, the subdivisions are obtained in the finest sub-clusters of the first stage based on another piercing threshold value Th\({}^{2}\) that can be formulated as:  
\[\text{fin}^{\text{i}}=\underset{\text{i}}{\text{argmax}}\Big{\{}\text{k}: \mathcal{D}_{\text{clu}^{\text{i}}}(\text{k})\geq\text{Th}^{2}\Big{\}} \tag{10}\]  
After the second subdivision stage, the third and final subdivision stage is initiated. Similar to the second stage, the finest sub-clusters are further split, and from that, another finest set of sub-clusters is computed based on Th\({}^{3}\). In this stage, there is a high chance of forming trivial sub-clusters within the sub-clusters with many elements.