# Improved Selective Deep-Learning-Based Clustering Ensemble
## 3 Improved Selective Deep Clustering Ensemble
### Selective Clustering Ensemble  
#### 3.2.1 Selective Strategy  
In the selective strategy, we firstly constructed the ensemble quality and diversity evaluation metrics of ensemble \(\Pi=\{\pi^{1},\pi^{2},\ldots,\pi^{S}\}\). \(\pi^{r}=\{C_{1}^{r},C_{2}^{r},\ldots,C_{n^{r}}^{r}\}\) denotes the \(r\)-th base clustering in \(\Pi\) and \(C_{n^{r}}^{r}\) indicates the \(n^{r}\)-th cluster of the \(r\)-th base clustering. \(n^{r}\) is the number of clusters.  
With regard to ensemble quality evaluation, for a given ensemble \(\Pi=\{\pi^{1},\pi^{2},\ldots,\pi^{S}\}\), we constructed a summational adjusted Rand index (SARI) as a means of evaluating the quality of each base clustering \(\pi^{r}\):  
\[SARI(\pi^{r},\Pi)=\sum_{i=1}^{S}\text{ARI}\Big{(}\pi^{r},\pi^{i}\Big{)} \tag{8}\]  
Here, the SARI measures the degree of consistency of the overall trend contained in a particular cluster \(\pi^{r}\) and ensemble \(\Pi\). Obviously, when the value of SARI is larger, the quality of \(\pi^{r}\) is higher. The ARI is the adjusted Rand index, which comprises the common external clustering metrics.  
With regard to ensemble diversity evaluation, for a given ensemble \(\Pi=\{\pi^{1},\pi^{2},\ldots,\pi^{S}\}\), we constructed pair-wise normalized mutual information (PNMI) to evaluate the diversity of the ensemble \(\Pi\):  
\[PNMI=\sum_{i\
eq j,\pi^{i},\pi^{j}\in\Pi}NMI\Big{(}\pi^{i},\pi^{i}\Big{)} \tag{9}\]  
NMI is the normalized mutual information, which is an external metric used in clustering to measure the degree of similarity between two clustering results. Here, when the value of PNMI is smaller, the diversity is richer in ensemble \(\Pi\).  
After defining the ensemble quality and diversity evaluation criteria, a certain number of higher quality and rich diversity candidate ensemble need to be selected from the ensemble \(\Pi=\{\pi^{1},\pi^{2},\ldots,\pi^{S}\}\). Three selection strategies can be used here:1. **Quality Strategy (QS)**. For a given ensemble \(\Pi=\{\pi^{1},\pi^{2},\ldots,\pi^{5}\}\), the QS adopts SARI\((\pi^{r},\Pi)\) to compute all the base clusterings in the ensemble \(\Pi\) and ranks them in descending order to select the top \(M\) (\(M\leq S\)) base clusterings with higher SARI values as the candidate ensemble. In general, base clusterings with higher SARI values show more overall trend consistency. Base clusterings with lower SARI values can be considered outliers in the ensemble and may be unfavorable for inclusion in the ensemble. 2. **Diversity Strategy (DS)**. The DS is a strategy that seeks to maximize ensemble diversity. For a given ensemble \(\Pi=\{\pi^{1},\pi^{2},\ldots,\pi^{5}\}\), we select \(M\) (\(M\leq S\)) base clusterings to minimize the PNMI. We can view this objective as a problem of finding weight-maximizing subgraphs, where the edge weight of each vertex is \(1-NMI\big{(}\pi^{i},\pi^{j}\big{)}\). However, this problem is NP-hard. Therefore, we approximate the solution of the problem using a simple greedy strategy. First, the highest-quality base clusterings are selected to form a new ensemble \(E\) using SARI computation, and then a base clustering from the ensemble \(\Pi\) is gradually selected to be added to the ensemble \(E\) so as to minimize the PNMI value. This process is repeated until the number of base clusterings in the ensemble \(E\) reaches \(M\). 3. **Balance Strategy (BS)**. The BS is a combination of the above two strategies. The two metrics SARI and PNMI are combined to form a new metric balance strategy index (BSI) as follows: \[BSI=\beta\sum_{i=1,2,\ldots,M}\text{SARI}\Big{(}\pi^{i},\Pi\Big{)}+(1-\beta) \sum_{i\
eq j}\Big{(}1-NMI\Big{(}\pi^{i},\pi^{j}\Big{)}\Big{)}\] (10) where \(\beta\) is an adjustment factor used to control for quality and diversity. Similarly, this joint metric BSI is solved greedily using a DS-like approach.