## 3 Tax Fraud Modeling with Neural Networks
### Dimension Adjustment: Reduction of the Dimension According to the Main Components
#### 3.3.1 Multilayer Perceptron Network Model Estimation and Diagnosis Phase
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Parameter Estimate**} \\ \hline \hline \multirow{3}{*}{**Predictor**} & \multicolumn{8}{c}{**Predicted**} \\ \cline{3-8}  & **H(1-D)** & **H(1-D)** & **H(1-D)** & **H(1-D)** & **H(1-D)** & **H(1-D)** & **H(1-D)** & **[marca = 0]** & **[marca = 1]** \\ \hline \multirow{2}{*}{**Input Layer**} & (Bias) & \(-0.274\) & \(1.639\) & \(-0.167\) & \(-0.122\) & \(-0.834\) & \(0.954\) & \(0.306\) \\  & FAC1\_1 & \(1.173\) & \(0.029\) & \(-0.794\) & \(-1.110\) & \(-0.988\) & \(-2.318\) & \(1.828\) \\  & FAC2\_1 & \(0.187\) & \(-0.319\) & \(0.649\) & \(-0.104\) & \(-0.278\) & \(0.508\) & \(0.430\) \\  & FAC3\_1 & \(-0.035\) & \(0.488\) & \(-0.210\) & \(-0.490\) & \(-0.437\) & \(0.713\) & \(-0.514\) \\  & FAC4\_1 & \(1.496\) & \(0.536\) & \(-1.207\) & \(-1.885\) & \(-0.466\) & \(0.200\) & \(1.700\) \\  & FAC5\_1 & \(-0.101\) & \(0.157\) & \(-0.052\) & \(0.284\) & \(-0.298\) & \(-0.330\) & \(-0.074\) \\  & FAC6\_1 & \(-0.098\) & \(0.467\) & \(-0.083\) & \(0.416\) & \(0.671\) & \(-0.892\) & \(0.898\) \\  & FAC7\_1 & \(4.657\) & \(-0.763\) & \(0.750\) & \(1.847\) & \(-0.314\) & \(0.072\) & \(2.417\) \\  & FAC8\_1 & \(0.289\) & \(0.889\) & \(-0.300\) & \(-0.342\) & \(-0.429\) & \(-0.245\) & \(-0.697\) \\  & FAC9\_1 & \(1.272\) & \(1.280\) & \(1.850\) & \(-1.144\) & \(1.084\) & \(0.195\) & \(-0.407\) \\  & FAC10\_1 & \(0.401\) & \(0.148\) & \(-0.022\) & \(0.181\) & \(-0.437\) & \(0.436\) & \(-0.833\) \\  & FAC1\_1 & \(-0.541\) & \(0.164\) & \(-1.281\) & \(0.343\) & \(0.087\) & \(0.682\) & \(0.767\) \\ \hline \hline \multicolumn{1}{l}{**Hidden Layer 1**} & (Bias) & & & & & \(-2.256\) & \(1.514\) \\  & H(1-1) & & & & & & \(-2.131\) & \(2.104\) \\  & H(1-2) & & & & & & \(-1.121\) & \(1.280\) \\  & H(1-3) & & & & & & \(6.633\) & \(-0.874\) \\  & H(1-4) & & & & & & \(1.102\) & \(1.594\) \\  & H(1-5) & & & & & &Figure 5 illustrates the structure of the neural network with the eleven nodes corresponding to the input or independent variables (main components), the sole hidden layer nodes labeled according to their synaptic weights, and an output node showing the two categories of the network model's dependent variable. The size of the input nodes indicates the magnitude of the effect of the corresponding independent variables on the dependent variable. Larger rectangles indicate a higher impact of the corresponding independent variable on the response. For example, the first, eighth, and fourth components have a greater effect on fraud. Be that as it may, the said effects will be numerically quantified later on. On the subject of the network model diagnosis, in the first place, it can be observed that the confusion matrix, in Table 4, presents high correct percentages, 84% of global percentages for the variable dependent of global fraud, for both training and testing of the predicted values. Figure 5: Neural network structure for the model estimation. Additionally, the graphical elements for diagnosis or robustness confirm the validity of the model. As can be seen in Figure 6, in the network's ROC curves representing tax fraud or tax compliance, both reflect a very high area between the curves and the diagonal (0.918), pointing to the network's very high predictive capacity. On the other hand, the gain curve reveals a bigger width between the two curves for high percentages of between 40% and 70%, which confirms that the greater the gain for the same percentage, the more accurate the prediction. Lastly, the lift chart also confirms the predictive capacity of the model, as the higher the percentage, the better the prediction made by the model.