# Understanding and Enhancement of Internal Clustering Validation Indexes for Categorical Data
## 4 Internal Clustering Validation Index: CUBAGE
### Cibage Index
\begin{tabular}{c c c c c c c c c c} \hline
**Dataset** & _CUBAGE_ & \(-F\) & \(-E\) & \(\textit{CUI}_{1\textit{k}}\) & _Clope_\({}_{1}\) & _Clope_\({}_{2}\) & _Clope_\({}_{3}\) & **R** \\ \hline Voting & (1) 0.443 & (8) 0.312 & (7) 0.324 & (1) 0.443 & (3) 0.418 & (3) 0.418 & (3) 0.418 & (6) 0.397 \\ Breast & (1) 0.674 & (6) 0.363 & (7) 0.358 & (1) 0.674 & (8) 0.016 & (4) 0.48 & (5) 0.407 & (3) 0.491 \\ Mushroom & (1) 0.458 & (5) 0.368 & (4) 0.391 & (3) 0.394 & (8) 0.196 & (2) 0.434 & (6) 0.365 & (7) 0.291 \\ Soybean & (1) 0.838 & (4) 0.746 & (3) 0.752 & (1) 0.838 & (8) 0.492 & (6) 0.553 & (6) 0.553 & (5) 0.572 \\ Car & (5) 0.05 & (2) 0.065 & (1) 0.072 & (3) 0.058 & (6) 0.026 & (6) 0.026 & (4) 0.054 \\ Heart & (1) 0.206 & (3) 0.175 & (4) 0.171 & (2) 0.205 & (8) 0.039 & (7) 0.165 & (6) 0.165 & (5) 0.167 \\ Dermatology & (1) 0.704 & (4) 0.647 & (3) 0.684 & (1) 0.704 & (8) 0.313 & (6) 0.362 & (5) 0.456 & (7) 0.321 \\ Zoo & (2) 0.808 & (3) 0.795 & (4) 0.783 & (5) 0.579 & (8) 0.479 & (7) 0.481 & (1) 0.823 & (6) 0.572 \\
**Average NMI \({}^{*}\)** & 0.522 & 0.434 & 0.442 & 0.487 & 0.247 & 0.365 & 0.402 & 0.358 \\
**Average Rank \({}^{*}\)** & 1.625 & 4.375 & 4.125 & 2.125 & 7.125 & 5.125 & 4.75 & 5.375 \\ \hline \end{tabular}  \({}^{*}\) Averaged over datasets (average values of each column). \end

### Discussion

We can see from the overall results that none of the internal CVIs consistently outperformed all of the other CVIs in either of the benchmark evaluation criteria over the eight datasets. This is because that the datasets are of different structures, and that the benchmark criteria NMI and ARI do not always agree with each other, especially when the quality of the partition is relatively low (see Figure 6c,e,f). However, we can observe that index _CUBAGE_ had better overall performance compared to others, from the perspectives of both the average scores and the averaged ranks. A detailed discussion follows:

Index _E_--the internal CVI without separation measure or assumption--performed well in the _k_-modes clustering when \(k\) was set to the actual number of classes. In the hierarchical clustering experiments, as we proved above, the value of \(E\) showed monotonicity with respect to the number of clusters, and the performance decreased notably. A similar performance drop appeared in the _k_-modes clustering when \(k\) was unknown. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Dataset** & _CUBAGE_ & \(-F\) & \(-E\) & _CUU\({}_{\text{LR}}\)_ & _Clope\({}_{\text{1}}\)_ & _Clope\({}_{\text{2}}\)_ & _Clope\({}_{\text{3}}\)_ & \(R\) \\ \hline Voting & (1) 0.443 & (5) 0.436 & (1) 0.443 & (1) 0.443 & (6) 0.376 & (6) 0.376 & (6) 0.376 & (4) 0.439 \\ Breast & (1) 0.674 & (4) 0.635 & (1) 0.674 & (1) 0.674 & (8) 0.015 & (7) 0.022 & (6) 0.434 & (5) 0.445 \\ Mushroom & (1) 0.458 & (5) 0.451 & (1) 0.458 & (1) 0.458 & (8) 0.037 & (1) 0.458 & (6) 0.435 & (7) 0.136 \\ Soybean & (1) 1 & (4) 0.977 & (1) 1 & (1) 1 & (7) 0.675 & (6) 0.692 & (5) 0.791 & (8) 0.672 \\ Car & (4) 0.048 & (1) 0.057 & (3) 0.05 & (2) 0.055 & (7) 0.038 & (8) 0.037 & (6) 0.042 & (5) 0.048 \\ Heart & (2) 0.181 & (3) 0.18 & (7) 0.171 & (5) 0.175 & (4) 0.177 & (8) 0.167 & (6) 0.173 & (1) 0.187 \\ Dermatology & (3) 0.742 & (4) 0.633 & (1) 0.744 & (2) 0.742 & (7) 0.505 & (6) 0.548 & (5) 0.582 & (8) 0.495 \\ Zoo & (1) 0.852 & (4) 0.818 & (3) 0.836 & (2) 0.838 & (5) 0.811 & (7) 0.804 & (6) 0.809 & (8) 0.781 \\
**Average NMI \({}^{*}\)** & 0.55 & 0.523 & 0.547 & 0.548 & 0.329 & 0.388 & 0.455 & 0.4 \\
**Average Rank \({}^{*}\)** & 1.75 & 3.75 & 2.25 & 1.875 & 6.5 & 6.125 & 5.75 & 5.75 \\ \hline \hline \end{tabular} \({}^{*}\) Averaged over datasets (average values of each column).

\end{table} 
Table 10: Average NMI values of the chosen _k_-modes partitions over 100 runs (_k_ fixed to the actual number of classes).