## 3 Methodology
### Collaborative Clustering
#### 3.2.4 Collaborative Step  
The local step clustering algorithm solutions matrix is then used in the initial stage of the collaborative step, during which the probability confusion matrices are calculated in order to generate the \(C(x,s)\) function. This idea was inspired by another work [28], in which the same confusion matrices are computed for a consensus-based method. The probability confusion matrix between two clustering algorithms is represented as shown in Equation (5). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathcal{X}\backslash\mathcal{A}\) & \(\mathcal{A}^{1}\) & \(\mathcal{A}^{2}\) & \(\mathcal{A}^{3}\) & \(\cdots\) & \(\mathcal{A}^{I}\) \\ \hline \(x_{1}\) & 0 & 1 & 0 & \(\cdots\) & 0 \\ \(x_{2}\) & 1 & 1 & 1 & \(\cdots\) & 1 \\ \(x_{3}\) & 0 & 1 & 0 & \(\cdots\) & 0 \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \(x_{N}\) & 0 & 0 & 1 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Solutions matrix. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Clustering Algorithm** & **Data Type** & **Description** \\ \hline K-means & Quantitative data & \
\begin{tabular}{c} Cluster similarity is measured based on the \\ mean value of the objects in the cluster. \\ \end{tabular} \\ \hline Gaussian mixture model (GMM) & Quantitative data & EM algorithm using Gaussian distribution. \\ \hline K-modes & Quantitative data &
\begin{tabular}{c} k-modes use a measure of difference based on \\ the number of mismatches between the modes \\ of the categorical variables. \\ \end{tabular} \\ \hline Bernoulli mixture model (BMM) & Quantitative data & EM algorithm using Bernoulli distribution. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Clustering Algorithms. \[\Psi^{i\to j}= \begin{pmatrix}\alpha_{1,1}^{i,j}&\cdots&\alpha_{1,k_{j}}^{i,j}\\ \vdots&\ddots&\vdots\\ \alpha_{1,k_{j}}^{i,j}&\cdots&\alpha_{k_{j},k_{j}}^{i,j}\end{pmatrix} ,\text{ where }\alpha_{k,j}^{ij}=\frac{\left|S_{k}^{i}\cap S_{l}^{j}\right|}{\left|S_{k}^{i} \right|} \tag{5}\]  
\(\Psi^{i\to j}\) is a matrix of size \(\mathcal{K}_{i}\times\mathcal{K}_{j}\) which maps the clusters of the \(\mathcal{A}^{i}\) algorithm to the clusters of the \(\mathcal{A}^{j}\) algorithm. Each element of the matrix \(\alpha_{k,l}^{i,j}\) represents the probability of placing the data in cluster \(l\) of the \(\mathcal{A}^{j}\) algorithm if this data were already in cluster \(k\) of the algorithm \(\mathcal{A}^{i}\).\(\left|\mathcal{S}_{k}^{i}\cap\mathcal{S}_{l}^{j}\right|\) represents the number of data points in cluster \(k\) of \(\mathcal{A}^{i}\) that are simultaneously in cluster \(l\) of \(\mathcal{A}^{j}\). / \(\left|\mathcal{S}_{k}^{i}\right|\) is the number of data points belonging to the \(k\) cluster of the \(\mathcal{A}^{i}\) algorithm. Once all the probability confusion matrices, \(\Psi\), have been computed, the second stage of the collaboration step commences. Given the matrices of \(\Psi\) and the results of other algorithms from the local step for a given clustering algorithm \(\mathcal{A}^{i}\), the consensus function \(C(x,s)\) to be estimated for cluster \(s\) becomes as shown in Equations (6) and (7). \[C(x,s)=\prod_{a_{j}\in\mathcal{A},j\
eq i}P\left(s\mid s_{x,a_{j}}\right) \tag{6}\]  
\(s_{x,a_{j}}\) is the cluster defined by algorithm \(a_{j}\) for object \(x\). At this juncture, other algorithm solutions are incorporated to estimate the probability of cluster \(s\). It is assumed that \(P\left(s\mid s_{x,a_{j}}\right)\) is independent. This assumption enables the use of the \(\Psi\) probability confusion matrices computed in the previous step, which yields Equation (7). \[C(x,s)=\prod_{a_{j}\in\mathcal{A},j\
eq i}\Psi^{j\to i}_{s_{x,a_{j}},s} \tag{7}\]  
Finally, Equation (3), which will be maximised during the collaborative step, becomes Equation (8):  
\[P(x\mid(s,\theta_{s}))P(s)=\frac{1}{Z}P(x\mid(s,\theta_{s}))\times C(x,s),\text { where }Z=J-1 \tag{8}\]  
where \(Z\) is a normalised constant independent of \(s\), and \(P(x\mid(s,\theta_{s}))\) is the local term.