## 3 Proposed Ensemble Clustering
### Problem Definition
#### 3.2.1 Production of Multiple Base Clustering Results
\[\Lambda_{ir}^{k}=\left\{\begin{array}{ll}1&\quad\left(\sum_{t=1}^{k-1}C_{t}+ j=p\right)\wedge\left(R_{i}^{\varphi_{ij}}=1\right)\wedge\left(CC_{pr}=1\right) \\ 0&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad





\[NMI(\pi^{*},\lambda)=\frac{2\sum_{i}\sum_{j}n_{ij}\ log\frac{n_{ij}n}{b_{i}b_{j}}}{ \sum_{ij}b_{j}\log\frac{b_{i}}{n}-\sum_{j}d_{j}\ log\frac{d_{j}}{n}} \tag{11}\]

The more similar the clustering result \(\pi^{*}\) (i.e., the consensus clustering result) and the ground truth clustering (the real labels of dataset), i.e., \(\lambda\), the more the value of their NMI (and ARI). ### Compared Methods

In order to investigate the performance of the proposed algorithm, we compare it with the state-of-the-art clustering ensemble algorithms including: (1) Evidence accumulation clustering (EAC) along with single link clustering algorithm as consensus function (EAC + SL) and (2) average link clustering algorithm as consensus function (EAC + AL) [9], (3) weighted connection triple (WCT) along with single link clustering algorithm as consensus function (WCT + SL) and (4) average link clustering algorithm as consensus function (WCT + AL) [27], (5) weighted triple quality (WTQ) along with single link clustering algorithm as consensus function (WTQ + SL) and (6) average link clustering algorithm as consensus function (WTQ + AL) [27], (7) combined similarity measure (CSM) along with single link clustering algorithm as consensus function (CSM + SL) and (8) average link clustering algorithm as consensus function (CSM + AL) [27], (9) cluster based similarity partitioning algorithm (CSPA) [29], (10) hyper-graph partitioning algorithm (HGPA) [29], (11) meta clustering algorithm (MCLA) [29], (12) selective unweighted voting (SUW) [17], (13) selective weighted voting (SWV) [17], (14) expectation maximization (EM) [37], and (15) iterative voting consensus (IVC) [40]. In addition, we compared the proposed method with other \"strong\" base clustering algorithms including: (1) the normal spectral clustering algorithm (NSC) [59], (2) \"density-based spatial clustering of applications with noise\" algorithm (DBSCAN) [62], (3) \"clustering by fast search and find of density peaks\" algorithm (CFSFDP) [63]. The purpose of this comparison is to test whether the proposed method is a \"strong\" clustering or not. ### Experimental Settings

A number of settings for the different ensemble clustering algorithms are listed in the following so as to ensure that they are reproducible. The number of clusters in each base clustering result is randomly set in the proposed ensemble clustering algorithm where the kmedoids clustering algorithm is also used to produce their basic clustering results.