# Consensus Big Data Clustering for Bayesian Mixture Models
## 4 Experimental Results  
In this section, we present the experimental results based on the methods from the preceding sections. Figure 1(a) shows the frequency histogram of \(y\) observations of the data, while Figure 1(b) shows the histogram of \(x\) and the density of the data.  
Figure 2(a) shows the MCMC sampling structure and the repetitive areas on \(y\) points over 300 iterations, while Figure 2(b) shows the posterior similarity matrix.  
For the experimental results, the following posterior parameters were utilized \(\mu_{1},\mu_{2},\mu_{3}\), \(\mu_{4}\), which represent the means of the posterior distributions for each of the four clusters; \(\sigma_{1}^{2},\sigma_{2}^{2},\sigma_{3}^{2},\sigma_{4}^{2}\) represent the variances of the posterior distributions for each of the four clusters; \(\eta_{1},\eta_{2},\eta_{3},\eta_{4}\) represent the proportions (or mixing weights) of the posterior distributions for each of the four clusters. The simulation parameters are the means \(\mu\) and variances \(\sigma^{2}\) of the clusters, as well as the mixing proportions \(\eta\) of the Poisson process mixture. The values  
Figure 2: Statistical sampling analysis of the PPM model.  
represent the estimated proportion of each of the four clusters in the PPM. The standard deviations give the uncertainty around these estimated proportions, with the lower and upper bounds indicating the credible interval. It is worth noting that the results presented in the table are just one possible output of the simulation study, and other simulations with different parameter settings may produce different results. Table 1 shows the results of a simulation study of PPM and consensus clustering.  
The clustering results of the proposed method in the MCMC framework are shown in Table 2. We conduct tests by adjusting the hyperparameters for different scenarios for which various numbers of clusters are produced, and their means and variances are calculated.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Par. Posterior** & **Mean** & **SD** & **Lower Bound** & **Upper Bound** \\ \hline \(\mu_{1}\) & 3.30 & 0.119 & 3.11 & 3.58 \\ \(\mu_{2}\) & 5.23 & 0.081 & 5.06 & 5.39 \\ \(\mu_{3}\) & 7.29 & 0.244 & 6.64 & 7.63 \\ \(\mu_{4}\) & 8.78 & 0.771 & 7.36 & 10.21 \\ \hline \(\sigma_{1}^{2}\) & 0.159 & 0.096 & 0.063 & 0.416 \\ \(\sigma_{2}^{2}\) & 0.312 & 0.081 & 0.181 & 0.498 \\ \(\sigma_{3}^{2}\) & 0.488 & 0.327 & 0.122 & 1.406 \\ \(\sigma_{4}^{2}\) & 2.82 & 1.128 & 1.022 & 5.229 \\ \hline \(\eta_{1}\) & 0.115 & 0.023 & 0.076 & 0.164 \\ \(\eta_{2}\) & 0.469 & 0.059 & 0.336 & 0.569 \\ \(\eta_{3}\) & 0.222 & 0.068 & 0.102 & 0.377 \\ \(\eta_{4}\) & 0.195 & 0.074 & 0.084 & 0.368 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results of PPM and consensus clustering.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test Scenario** & **Hyperparameters** & **No. of Clusters** & **Mean** & **Variance** \\ \hline
1 & \(\alpha=1.0,m=0.5\) & 4 & 1.25 & 0.12 \\
2 & \(\alpha=1.2,m=0.6\) & 5 & 1.10 & 0.09 \\
3 & \(\alpha=1.5,m=0.7\) & 3 & 1.45 & 0.15 \\
4 & \(\alpha=0.9,m=0.4\) & 6 & 1.05 & 0.08 \\
5 & \(\alpha=1.3,m=0.8\) & 2 & 1.60 & 0.17 \\
6 & \(\alpha=1.1,m=0.3\) & 7 & 0.95 & 0.07 \\
7 & \(\alpha=1.4,m=0.9\) & 4 & 1.30 & 0.11 \\
8 & \(\alpha=1.6,m=0.2\) & 5 & 1.15 & 0.10 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental results using the proposed Bayesian nonparametric clustering in MCMC framework.  
Figure 3: Statistical sampling analysis of the PPM and consensus clustering.