## 3 Methodology
### Collaborative Clustering
#### 3.2.1 Formalism  
In horizontal collaborative clustering, a limited set of algorithms, \(\mathcal{A}=\left\{\mathcal{A}^{1},\mathcal{A}^{2},\cdots,\mathcal{A}^{l}\right\}\), is considered, which operates on the same data items, albeit with access to different features, and possibly also looking for a different number of clusters. Let \(\mathcal{X}=\left\{x_{1},x_{2},\cdots,x_{N}\right\}\), and \(x_{n}\in\mathbb{R}^{d}\) be a data set containing \(\mathcal{N}\) elements, each with real numeric properties.  
Each \(\mathcal{A}^{i}\) clustering algorithm has parameters, \(\theta^{i}\), to describe the clusters or model, and produces its clustering solution, \(\mathcal{S}^{i}\), made of \(\mathcal{K}_{i}\) clusters based on the features of the \(\mathcal{X}^{i}\subseteq\mathcal{X}\) data set to which it has access. In the case of hard clustering, \(\mathcal{S}^{i}\) can be translated into a solution vector of size \(\mathcal{N}\), soft clustering into a matrix of size, \(\mathcal{N}\times\mathcal{K}_{i}\). This matrix is denoted \(\mathcal{S}^{i}=\left(s^{i}{}_{n,c}\right)\), where \(1\leq n\leq\mathcal{N}\) and \(1\leq c\leq\mathcal{K}_{i}\). Thus, the \(\mathcal{S}^{i}\) solutions generated by the algorithms are 2D matrices of size \(\mathcal{N}\times\mathcal{K}_{i}\), where each element \(\left(s^{i}{}_{n,c}\right)\) expresses the  
\begin{table}
\begin{tabular}{c c c} \hline
**Feature** & **Data Type** & **Description** \\ \hline Age & Quantitative & age of the patient (years) \\ Sex & Qualitative & sex of the patient (M: male, F: female) \\ Chest Pain Type & Qualitative & chest pain type (TA: typical angina, ATA: atypical angina, \\  & & NAP: non-anginal pain, ASY: asymptomatic) \\ RestingBP & Quantitative & resting blood pressure (mm Hg) \\ Cholesterol & Quantitative & serum cholesterol (mm/dL) \\ FastingBS & Qualitative & fasting blood sugar (1: if FastingBS? 120 mg/dL, 0: otherwise) \\  & & resting electrocardiogram results (normal: normal, ST: having ST-T wave \\ RestingECG & Qualitative & abnormality (T wave inversions and ST elevation or depression of and \\  & & 0.05 mV), LVH: showing probable or definite left ventricular hypertrophy \\  & & by Estes’ criteria) \\ MaxHR & Quantitative & maximum heart rate achieved (numeric value between 60 and 202) \\ ExerciseAngina & Qualitative & exercise-induced angina (Y: yes, N: no) \\ Oldpeak & Quantitative & oldpeak = ST (numeric value measured in depression) \\ ST\_Slope & Qualitative & the slope of the peak exercise ST segment (Up: upsloping, Flat: flat, \\  & & Down: downsloping) \\ HeartDisease & Qualitative & output class (1: heart disease, 0: normal) \\ \hline \end{tabular}
\end{table}
Table 1: Description of data set.  
responsibility (probability) given by the algorithm \(\mathcal{A}^{i}\) to cluster \(c\) of the \(x^{i}\ _{n}\) data element. This matrix is transformed into a solution vector of size \(\mathcal{N}\) by adopting the fact that each data set record belongs to one set with the highest probability, \(\mathcal{S}^{i}=\text{arg}\max(s^{i}_{n,c})\).  
The method assumes that any clustering algorithm whose model is collaboratively optimised attempts to optimise an objective function similar to Equation (2):  
\[\mathcal{S}=\underset{\mathcal{S}}{\text{argmax}}(P(\mathcal{S}\ |\ (\mathcal{X}, \theta)))=\underset{\mathcal{S}}{\text{argmax}}(P(\mathcal{X}\ |\ (\mathcal{S}, \theta))\times P(\mathcal{S})) \tag{2}\]  
Equation (2) can be solved using a local minimisation process, as expressed in Equation (3):  
\[P(\mathcal{X}\ |\ (\mathcal{S},\theta))P(S)=\prod_{t}^{\mathcal{N}}P(x_{t}\ |\ (s_{t}, \theta_{s_{t}}))\times P(s_{t}) \tag{3}\]  
Each algorithm may be based on different statistical models, e.g., Gaussian or polynomial, amongst others. It is hypothesised that this collaborative method can only be applied to algorithms attempting to optimise an equation similar to Equation (2). Most of the symbols used in this section are summarised in Table 2.