## 3 Proposed Method
### Simulation with Artificial Examples
#### 3.5.1 Artificial Example I: Variable Investigation
The asterisk indicates extremely influential variable module(s). For each variable module, we also present the AUC value (see Section 4.4 for a detailed discussion of AUC values) assuming a classifier is built using this variable module alone. The detailed variable modules and their corresponding I-scores are exhibited in Table 3. In this table, we present four columns: New Modules, Variables, Associated I-score, and Corresponding AUC. The new module is constructed using (6). The variables were obtained from the BDA. The I-score of a variable set is computed using Equation (2). For computation of the AUC, please see Section 4.4 In the artificial data set that has \(6\times 6=36\) variables, a window size of \(2\times 2\) allows us to generate \((6-2+1)^{2}=25\) variable modules. Each module is created using the selected variables from a local \(2\times 2\) window after using the BDA, and the module is generated using Formula (4). We observe that the variable module \(X_{1}^{\dagger}\) that is formed based on \(\{X_{1},X_{2}\}\) is the most influential candidate because it directly links to predictivity. This can be confirmed using the AUC value if this module is used to build a classifier by itself. A different window size can be selected. In this artificial example, we attempt \(3\times 3\). For data with \(6\times 6=36\) variables, a window size of \(3\times 3\) allows us to create \((6-3+1)^{2}=16\) variable modules. The results of this experiment are listed in Table 4. In Table 4, we observe a slightly different set of variable modules selected. We notice that the most influential variable module \(X_{1}^{\dagger}\) that is based on \(\{X_{1},X_{2}\}\) remains the same. In other words, if a local section of the data has real predictive power, the proposed method will almost always identify the information. This can be verified using the AUC value associated with the variable module. Since we pass a \(3\times 3\) window, we are able to sometimes screen for higher-order interactions. As discussed above, we have the I-score presented in Table 4, so that feature ranking is possible if desired. From this table, we observe that another module \(X_{9}^{\dagger}\), that was constructed using \(\{X_{3},X_{4},X_{5}\}\), is also important. Though it has a lower I-score than the first module \(X_{1}^{\dagger}\), it has a higher I-score than any other candidates in the data set. **Prediction. ** After the variable modules (the variables named \(X^{\dagger}\)'s) are constructed, we can proceed with building the neural network classifier. In Table 3, there are 25 variable modules, i.e., \(\{X_{1}^{\dagger},\ldots,X_{25}^{\dagger}\}\). We use these variable modules as the input layer of a neural network architecture. In other words, the input layer has 25 neurons (these neurons are the exact 25 variable modules named \(X^{\dagger}\)'s). For the variable modules in Table 4, the input layer consists of 16 neurons, which are \(\{X_{1}^{\dagger},\ldots,X_{16}^{\dagger}\}\). Since we are dealing with a rather small input layer (only 25), no hidden layer is recommended. The output layer can simply be one neuron, and the final outcome \(\hat{Y}\) can be computed using a sigmoid function as follows:  
\[\hat{Y}:=1/(1+\exp(-\sum_{j}^{25}(w_{j}X_{j}^{\dagger})) \tag{10}\]