# On the Influence of Data Imbalance on Supervised Gaussian Mixture Models
## 5 Results
### Wine Quality Data  
We consider the wine quality data [22] available from the UCI Machine Learning Data Repository [23]. The dataset contains data on 4898 white variants of the Portuguese \"Vinho Verde\" wine. For each wine, the results from 11 physicochemical tests are available, such as fixed, volatile, and citric acidity, residual sugar, chlorides, free and total sulfur dioxide, density, pH, sulphates, and alcohol. Moreover, the quality of each wine was assessed based on sensory data, with scores ranging from 0 to 10. For the present analysis, wines have been classified into two groups: those of High quality (with a score of at least 8) and those classified as MedLow quality (with a score less than 8). Notably, only 3.67% of the wines achieved the distinction of high quality, leading to a clear imbalance between the classes. The dataset was partitioned into a training set, which comprised approximately 2/3 of the wines, and a test set that encompassed the remaining 1/3 of total observations. Given that the proportion of the minority class is 0.0367, the imbalance ratio in the training set is equal to \(\mathrm{IR}=26.22\). Figure 3 illustrates the results derived from applying the models presented in Section 2.2, along with the remedies discussed in Section 4, to the test set. Without any adjustment for correcting class imbalance both the EDDA model and the McustDA model have a quite low TPR, which leads to low values for recall and sensitivity. The precision and specificity are higher, but this does not compensate for the deficiency in detecting the minority class. Overall, both F2 and Balanced Accuracy are smaller for the unadjusted models that use the initial partition of the data. Cost-sensitive predictions, as discussed in Section 4.1, employ the imbalance ratio mentioned above for both EDDA and McustDA models. Recall that this is equivalent to use the proportion of the minority class as the classification threshold. Instead, optimal threshold for the EDDA model estimated by cross-validation is \(\zeta_{\mathrm{CV}}=0.08\), while using the adjusted prior probabilities, as discussed in Section 4.2, it is recalibrated as \(\zeta_{\mathrm{ADJ}}=0.299\). At the same time, for the McustDA model the optimal threshold is estimated by cross-validation as \(\zeta_{\mathrm{CV}}=0.01\), while the adjusted threshold is estimated as \(\zeta_{\mathrm{ADJ}}=0.0501\). As \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Method** & **Precision/** & **Recall/** & **F2** & **Sensitivity/** & **Specificity/** & **Balanced** \\  & **PPV** & **TPR** & & **TPR** & **TNR** & **Accuracy** \\ \hline \hline MCLUSTDA & 0.8550 (0.0009) & 0.6119 (0.0202) & 0.6487 (0.0019) & 0.6119 (0.0022) & 0.9885 (0.0001) & 0.8002 (0.0011) \\ \hline MCLUSTDA+costSens & 0.5110 (0.0016) & 0.9959 (0.0014) & 0.8159 (0.0005) & 0.9956 (0.0014) & 0.8978 (0.0008) & 0.9287 (0.0004) \\ \hline MCLUSTDA+adjPriorThreshold & 0.5134 (0.0015) & 0.9598 (0.0012) & 0.8166 (0.0006) & 0.9589 (0.0012) & 0.8989 (0.0006) & 0.9289 (0.0004) \\ \hline MCLUSTDA+opTThreshold & 0.4979 (0.0036) & 0.9679 (0.0017) & 0.8129 (0.0013) & 0.9679 (0.0017) & 0.8903 (0.0017) & 0.9291 (0.0004) \\ \hline MCLUSTDA+downSamp & 0.5043 (0.0016) & 0.9596 (0.0016) & 0.8113 (0.0007) & 0.9576 (0.0016) & 0.8952 (0.0005) & 0.9264 (0.0005) \\ \hline MCLUSTDA+optsemp & 0.5136 (0.0015) & 0.8717 (0.0028) & 0.7648 (0.0019) & 0.8717 (0.0028) & 0.9082 (0.0005) & 0.8900 (0.0013) \\ \hline MCLUSTDA+smote & 0.4946 (0.0013) & 0.8108 (0.0027) & 0.7187 (0.0019) & 0.8108 (0.0027) & 0.9079 (0.0005) & 0.8893 (0.0013) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._shown in Figure 3, both the F2 score and balanced accuracy exhibit higher values when either the threshold-based approaches or the sampling-based approaches are employed to address class imbalance. While MclustDA models exhibit slightly better results compared to EDDA models, the variation among different methods for addressing imbalance is negligible.