## 3 Methodology
### Collaborative Clustering
#### 3.2.2 Problem Formulation  
The proposed method enhances the clustering process by sharing information between several results obtained from different clustering algorithms. The originality of the proposed approach is that the collaboration step can use the clustering results obtained from any algorithm during the local step. The main issue comes from Equation (3) in calculating \(P(s)\), which is the probability of each cluster. This is not known in advance as most clustering algorithms include an assumption to calculate \(P(s)\). For example, the K-means algorithm and some versions of the EM algorithm assume that all combinations have the same occurrence probability. Most unsupervised probabilistic classifiers follow this method. The principal concept is to measure the event of all clusters after each iteration. The local probability hypothesis, primarily used in computer vision, will depend on the neighbour composition of the observed data, rather than measuring the global \(P(s)\) of the entire data set, \(P(s)\). Thus, its value will be determined based on the clusters assigned to the neighbour data.  
In the proposed collaborative method, a hypothesis similar to the local hypothesis was followed. During the collaborative step, it will be considered that the probability of occurrence of each \(P(s)\) cluster will not be a global probability. Instead, \(P(s)\) will be  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Development** & **Comment** \\ \hline \(\mathcal{X}^{i}\) & \(\mathcal{X}^{i}=\left\{x^{i},\ _{1},x^{i}\ _{2},\cdots,x^{i}\ _{N}\right\},x^{i}\ _{n}\in \mathbb{R}^{d}\) & The subset of the data observed by algorithm \(\mathcal{A}^{i}\). \\ \hline \(\mathcal{X}\) & \(\mathcal{X}=\{\mathcal{X}^{1},\mathcal{X}^{2},\cdots,\mathcal{X}^{l}\}\) & All data with all views. \\ \hline \(\mathcal{S}^{i}\) & \(\mathcal{S}^{i}=\left\{S^{i}_{1},S^{i}_{2},\cdots,S^{i}_{N}\right\},S^{i}_{n} \in\mathbb{R}^{k^{i}}\) & The solution vector of the algorithm \(\mathcal{A}^{i}\). \\ \hline \(\mathcal{S}\) & \(\mathcal{S}=\{\mathcal{S}^{1},\mathcal{S}^{2},\cdots,\mathcal{S}^{l}\}\) & Solution vectors for all clustering algorithms. \\ \hline \(\mathcal{A}^{i}\) & \(\mathcal{A}^{i}=\left\{\mathcal{X}^{i},\mathcal{X}^{i},\theta^{i},\mathcal{S}^{i }\right\}\) & The set of distribution parameters for all algorithms. \\ \hline \multirow{2}{*}{\(\mathcal{Z}\)} & An algorithm looking for \(\mathcal{K}_{i}\) clusters of distribution & \\  & parameters \(\theta^{i}\) in the subset \(\mathcal{X}^{i}\) and finding & \\  & a solution vector \(\mathcal{S}^{i}\) & \\ \hline \(\Psi^{i\to j}\) & \(\Psi^{i\to j}=\left\{\Psi^{1\to j},\Psi^{2\to j},\cdots,\Psi^{2}, \cdots,\theta^{l}\right\}\) & The normalised constant is independent of \(s\). \\ \hline \(a^{i,j}_{k,l}\) & \(a^{i,j}_{k,l}\in\Psi^{i\to j}\) & The consensus matrices map the other algorithms’ \\  & & clusters to the ones of the current algorithm \(\mathcal{A}^{l}\). \\ \hline \hline \end{tabular}
\end{table}
Table 2: Notation.  
associated with clustering choices made by other algorithms for the same data point. The objective is to establish a method of modifying Equation (3) such that the solutions of all the clustering algorithms are used to compute \(P(s)\), as indicated by Equation (4).  
\[P(\mathcal{X}\mid(\mathcal{S},\theta))P(S)=\prod_{t}^{\mathcal{N}}P(x_{t}\mid(s _{t},\theta_{s_{t}}))\times C(x_{t},s_{t}) \tag{4}\]  
\(C(x_{t},s_{t})\) is a function that determines the value of \(P(s)\) by the consensus of all other clustering algorithms for the same data point \(x_{t}\).