## 3 Proposed Ensemble Clustering
### Notifications and Definitions
#### 3.1.1 Clustering  
A set of \(C\) non-overlapping subsets of data set can be called as a clustering result (or abbreviately a clustering) or a partitioning result (or abbreviately a partitioning), if the subsets union is the entire data set and the intersection of each pair of subsets is null; any subset of a data set is called a cluster. A clustering is shown by \(\Phi\), a binary matrix, where \(\Phi_{:i}\), a vector of size \(|D_{:1}|\), represents the \(i\)-th cluster; and \(\Phi_{:i}^{T}\), a vector of size \(C\), represents which cluster the \(i\)-th data point belongs to. Obviously, \(\sum_{j=1}^{C}\Phi_{ij}=1\) for any \(i\) in \(\{1,2,\ldots,|D_{:1}|\}\); and \(\sum_{i=1}^{|D_{:1}|}\Phi_{ij}>0\) for any \(j\) in \(\{1,2,\ldots,C\}\); and also we have \(\sum_{i=1}^{|D_{:1}|}\sum_{j=1}^{C}\Phi_{ij}=|D_{:1}|\). The center of each cluster \(\Phi_{:i}\) is a data point shown by \(M^{\Phi_{:i}}\), and its \(j\)-th feature is defined as Equation (1) [51].  
\[M_{j}^{\Phi_{:i}}=\frac{\sum_{k=1}^{|D_{:1}|}\Phi_{ki}D_{kj}}{\sum_{k=1}^{|D_{: 1}|}\Phi_{ki}} \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(D\) & A dataset \\ \(D_{i:}\) & \(i\)-th data object in dataset \(D\) \\ \(L_{D_{:}}\) & Real label of the \(i\)-th data object in dataset \(D\) \\ \(D_{ij}\) & \(j\)-th feature from \(i\)-th data object \\ \(|D_{:1}|\) & The size of data set \(D\) \\ \(|D_{1:1}|\) & The number of features in data set \(D\) \\ \(\mathbf{\Phi}\) & A set of initial clustering results \\ \(\Phi^{i}\) & \(i\)-th clustering result in the ensemble clustering \(\mathbf{\Phi}\) \\ \(\Phi_{:k}^{i}\) & \(k\)-th cluster in the \(i\)-th clustering result of the ensemble clustering \(\mathbf{\Phi}\) \\ \(\Phi_{jk}^{i}\) & A Boolean indicating whether \(j\)th data point of the given dataset belongs to the \(k\)-th cluster in the \\ \(i\)-th clustering result of the ensemble clustering \(\mathbf{\Phi}\) or not \\ \(C\) & The number of the consensus clusters in given dataset \\ \(R^{\Phi_{:k}^{i}}\) & A valid sub-cluster from the cluster \(\Phi_{:k}^{i}\) \\ \(R_{j}^{\varphi_{:k}}\) & A Boolean indicating whether \(j\)th data point of the given dataset belongs to the valid sub-cluster \\  & of the \(k\)-th cluster in the \(i\)-th clustering result of the ensemble clustering \(\mathbf{\Phi}\) or not \\ \(\gamma\) & The neighboring radius parameter of the valid cluster in the proposed algorithm \\ \(M^{\Phi_{:i}}\) & center point of cluster \(\Phi_{:i}\) \\ \(M^{\Phi_{:i}}_{w}\) & \(w\)-th feature from center point of cluster \(\Phi_{:i}\) \\ \(\pi^{*}\) & Consensus clustering result \\ \(sim(u,v)\) & Similarity between two clusters \(u\) and \(v\) \\ \(T_{q}(u,v)\) & \(q\)-th hypothetical cluster between two center points of clusters \(u\) and \(v\) \\ \(p_{q}(\pi^{i},\pi\prime)\) & The center of \(q\)-th hypothetical cluster between two clusters \(u\) and \(v\) \\ \(B\) & The size of the ensemble clustering \(\mathbf{\Phi}\) \\ \(C_{i}\) & The number of the clusters in the \(i\)-th clustering result \(\Phi^{i}\) \\ \(G(\mathbf{\Phi})\) & The graph defined on the ensemble clustering \(\mathbf{\Phi}\) \\ \(V(\mathbf{\Phi})\) & The nodes of the graph defined on the ensemble clustering \(\mathbf{\Phi}\) \\ \(E(\mathbf{\Phi})\) & The edges of the graph defined on the ensemble clustering \(\mathbf{\Phi}\) \\ \(\lambda\) & A clustering result similar to the real labels \\ \hline \hline \end{tabular}
\end{table}
Table 1: The used notifications and symbols.