# How to Open a Black Box Classifier for Tabular Data
## 3 Results
### Synthetic Data
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**AUC [CI]**} & \multicolumn{1}{c}{**No. Input**} & \multicolumn{1}{c}{**Training**} & \multicolumn{1}{c}{**Optimisation**} & \multicolumn{1}{c}{**Performance**} \\  & **Variables** & **(n = 6000)** & **(n = 2000)** & **Estimation (n = 2000)** \\ \hline
**Optimal classifier** & **3** & **0.840 [0.822,0.859]** & **0.817 [0.783,0.851]** & **0.836 [0.805,0.868]** \\ MLP & 9 & 0.840 [0.822,0.859] & 0.809 [0.775,0.843] & 0.832 [0.801,0.864] \\ _SVM_ & 9 & _0.797 [0.779,0.815]_ & _0.764 [0.729,0.798]_ & _0.786 [0.755,0.817]_ \\ GBM & 9 & 0.831 [0.816,0.847] & 0.796 [0.767,0.826] & 0.813 [0.786,0.840] \\ \hline
**_PRiSM models_** & _Components_** & & _Dirac measure_ \\ Lasso & 3 & 0.837 [0.818,0.855] & 0.811 [0.777,0.845] & 0.821 [0.797,0.861] \\ PRN & 3 & 0.837 [0.819,0.856] & 0.812 [0.778,0.846] & 0.830 [0.799,0.862] \\ PRN–Lasso & 3 & 0.837 [0.819,0.856] & 0.812 [0.778,0.846] & 0.831 [0.799,0.862] \\ prSVM & 6 & 0.808 [0.792,0.824] & 0.776 [0.745,0.808] & 0.805 [0.777,0.833] \\ prGBM & 4 & 0.825 [0.809,0.841] & 0.798 [0.768,0.828] & 0.810 [0.781,0.839] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Classification performance for the three-way interaction measured by the AUC [CI]. Three input variables are involved, \(x_{7}\), \(x_{8}\), and \(x_{9}\). Figure 3: Class allocation for the 2-circle synthetic data set as a function of \(x_{1}\) and \(x_{2}\) showing: (**a**) The stochastic class labels; and (**b**) The correct classes that are used to find the optimal AUC. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**AUC [CI]**} & \multicolumn{1}{c}{**No. Input**} & \multicolumn{1}{c}{**Training**} & \multicolumn{1}{c}{**Optimisation**} & \multicolumn{1}{c}{**Performance**} \\  & **Variables** & **(n = 6000)** & **(n = 2000)** & **Estimation (n = 2000)** \\ \hline
**_PRiSM models_** & _Components_ & & _Lebesgue measure_ & \\ Lasso & 3 & 0.815 [0.801,0.828] & 0.832 [0.808,0.856] & 0.813 [0.789,0.837] \\ PRN & 3 & 0.816 [0.802,0.829] & 0.835 [0.811,0.858] & 0.814 [0.790,0.838] \\ PRN–Lasso & 3 & 0.816 [0.802,0.830] & 0.835 [0.811,0.858] & 0.815 [0.791,0.839] \\ prSVM & 4 & 0.799 [0.786,0.812] & 0.812 [0.790,0.834] & 0.796 [0.773,0.819] \\ prGBM & 8 & 0.817 [0.805,0.829] & 0.828 [0.808,0.849] & 0.810 [0.789,0.831] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Figure 4: Class allocation for the XOR data set as a function of \(\mathrm{x_{3}}\) and \(\mathrm{x_{4}}\) showing: **(a)** The stochastic class labels; **(b)** The correct classes used to find the optimal AUC; **(c)** The two-way interaction term identified by the Dirac measure; and **(d)** The interaction estimated with the Lebesgue measure, which is almost identical to the curve in **(c)**. Both surfaces are the only terms in the GAM, and closely correspond to the logit of the ideal XOR prediction surface. The main difference to theory is that the values at the four corners that saturate at finite values, whereas in theory, they extend to infinity in both vertical directions. This, however, has little impact on the crucial region for classification, which is the class boundary. Figure 5: Class allocation for the synthetic data set representing the logical AND a function of \(\mathrm{x_{5}}\) and \(\mathrm{x_{6}}\) showing: **(a)** The stochastic class labels; and **(b)** The correct classes to find the optimal AUC. The MLP-derived PRiSM models identified the correct ANOVA components for all data sets, with both the Dirac and Lebesgue measures. The three-way interaction term is a product of three inputs, not a pure interaction term. Therefore, its decision boundary can be approximated even in the absence of a third-order term, with three univariate partial responses sufficient to get close to the optimal prediction accuracy. Note that the predicted response in Figure 4c,d for the XOR task is very close to the bilinear surface corresponding to the theoretically correct response given by multilinear algebra ([PERSON], 2000).