# How to Open a Black Box Classifier for Tabular Data
## 3 Results
### Real-World Data
The final partial responses derived at the second application gradient descent (solid lines) are shown alongside the partial responses from the original MLP (dashed lines). Five covariates are represented, namely (**a**) Pregnancies, (**b**) Glucose, (**c**) BMI, (**d**) DPF and (**e**) Age. Figure 7: As for Figure 6, with the Lebesgue measure, the component functions of the GAM are very similar for both measures. They have a similar structure and range of contributions to the logit. Despite being fitted with a generic non-linear model, the MLP, several of the partial responses are linear. Variable “DPF” shows a saturation effect, as might be expected, while the log odds of “Age” as an independent effect peak around the age of 40. Note that data sparseness for higher values will result in greater uncertainty in the estimation of the partial response. The same size covariates are represented (**a–e**) as in Figure 6.  
Figure 8: Partial responses for the German Credit Card data set, using the same notation as the previous figures. Four univariate responses and a bivariate response are shown namely for the covariates (**a**) Status of checking account, **(b**) Duration of loan, (**c**) Credit history and (**d**) Credit amount, together with (**e**) the pairwise interaction between Credit amount and Duration of loan. Figure 9: As for Figure 8, with the Lebesgue measure for the same covariates in (**a**–**d**), but with two pairwise interactions involving the variables listed in (**e**,**f**). Despite the different nature of the two measures, they offer entirely consistent interpretations, with the only difference being the selection by the Lasso model of a second bivariate interaction term, albeit with a range in contribution to the logit that is five times smaller than for the interaction term involving “Credit amount” and “Duration”. Among the seven covariates in the Diabetes data set, five occurred together as univariate responses in all of the random initialisations for the MLP-Lasso, PRN-Lasso, and the two measures. They are \"Pregnancies\", \"Glucose\", \"BMI\", \"DPF\", and \"Age\". An interaction term involving \"Glucose\" and \"DPF\" was present in three random initialisations. The set of models obtained is, therefore, remarkably stable. The partial responses for the recurrent univariate effects are shown in Figures 6 and 7. The German Credit Card data set is more challenging. Out of 24 variables, six were present in all initialisations for both measures: \"Duration\", \"Credit history\", \"Savings accounts\", \"Period of employment\", and the two variables labelled x\({}_{16}\) and x\({}_{17}\). In the case of the Lebesgue measure, three more variables recurred in all 10 initialisations, namely \"Status of checking account\", \"Other instalment plans\", and \"Worker status\". In addition, the variable \"Credit amount\" featured as a univariate or a bivariate term in eight initialisations. These ten variables were selected to obtain the models for which a selection of component functions is shown in Figures 8 and 9. Figure 10: Nomogram of the PRN–Lasso model obtained for the Statlog Shuttle data set using the Dirac measure with a training/test split of n = 43,500 and 14,500, respectively: (**a**) Shows the raw data for the two variables selected, which corresponds well with two partial responses in the final model, namely: (**b**) shows the main effect involving x\({}_{9}\); and (**c**) plots the two-way interaction between the two variables in the model, x\({}_{1}\) and x\({}_{9}\). Four univariate functions for multi-valued input variables are consistently monotonically decreasing and very close to linear, suggesting that these indicators are well calibrated as independent effects on credit risk, quantifying reductions in risk with rising input values. \"Duration\" shows saturation in its contribution to risk for large values, and \"Credit amount\" has a non-linear response with a minimum value.