## 3 Proposed Method
### Simulation with Artificial Examples
#### 3.5.2 Artificial Example II: Four Modules with Discrete Variables  
For a more complex example, we designed an experiment that is similar to the one before but with more high-order interactions. We create a data set with 49 variables, \(\{X_{i}:1\leq i\leq 49\}\). Each \(X_{i}\) takes binary form, 0 or 1. We define dependent variable \(Y\) as the following form, namely Model (12),  
\[Y=\left\{ \begin{array}{ll}X_{1}+X_{2}&(\text{mod 2})\text{ w/ prob. }\frac{1}{2}\\ X_{2}+X_{3}+X_{4}&(\text{mod 2})\text{ w/ prob. }\frac{1}{2}\\ X_{6}\cdot X_{7}&(\text{mod 2})\text{ w/ prob. }\frac{1}{2}\\ X_{8}\cdot X_{9}\cdot X_{10}&(\text{mod 2})\text{ w/ prob. }\frac{1}{8}\\ \end{array} \right. \tag{12}\]  
which is a more complicated problem than Model (7). The complexity is higher since there are four modules instead of two modules in Model (12). Figure 5: The above architecture presents a feed forward neural network with three input variables. The input variables are \(\{X^{\dagger}_{1},X^{\dagger}_{2},X^{\dagger}_{3}\}\), which are variable modules created using Equation (6). **Scenario I. ** Suppose we know the model formulation. We see that the first module occurs with a probability of 0.5. In this case, by correctly identifying the first module, we achieve a 50% accuracy rate. Additionally, this module is able to determine half of the rest of the occurrences in the data. Hence, the theoretical [PERSON] rate for model is 75%, which is computed below. Consider the first correct module identified as \(S_{1}=\{X_{1},X_{2}\}\). In this case, the theoretical prediction rate \(\theta_{c}(S_{1})\) can be calculated, assuming \(n\) samples in the training data,  
\[ \begin{array}{rcl}\theta_{c}(S_{1})&=&\theta_{c}(\{X_{1},X_{2}\})\\ &=&\frac{1}{\eta}\sum_{i}^{n}\mathds{1}(\hat{Y}=Y)\\ &=&\frac{1}{\eta}\sum_{i}^{n}\mathds{1}(\underbrace{(X_{1}+X_{2})}_{(\text{ mod 2})}=Y)\\ &=&50\%+50\%\cdot 50\%\\ &=&75\%\end{array}  \tag{13}\]  
The second correct module, though informative, occurs less frequently than the first module. Consider this second variable module \(S_{2}=\{X_{2},X_{3},X_{4}\}\), then we can compute the theoretical prediction rate \(\theta_{c}(S_{2})\) as the following:  
\[ \begin{array}{rcl}\theta_{c}(S_{2})&=&\theta_{c}(\{X_{2},X_{3},X_{4}\})\\ &=&\frac{1}{\eta}\sum_{i}^{n}\mathds{1}(\hat{Y}=Y)\\ &=&\frac{1}{\eta}\sum_{i}^{n}\mathds{1}(\underbrace{(X_{2}+X_{3}+X_{4})}_{( \text{mod 2})}=Y)\\ &=&25\%+\underbrace{(1-25\%)\cdot 50\%}_{\text{remains}}=62.5\%\end{array}  \tag{14}\]  
which means the correct identification of the second variable module should be 62.5%. In this case, this second variable module \(\{X_{2},X_{3},X_{4}\}\) together occurs 25%, so these appearances \(S_{2}\) should correctly make the prediction. For the remaining observations that occur \(1-25\%=75\%\) of the times, the second variable module only makes correct guesses half of the time. Hence, this computation takes the form of Equation (14), which results in 62.5%. \begin{table}
\begin{tabular}{l c} \hline \hline
**Algorithm** & **Test AUC** \\ \hline Theoretical Prediction & 0.75 \\ \hline Net-3 & 0.50 \\ LeNet-5 & 0.50 \\ \hline Interaction-based Conv. Layer: & \\ Window size: \(2\times 2\) & \\ (25 modules listed in Table 3) & \\ Using \(\hat{Y}\) defined in Equation (10) & \\ Interaction-based Conv. Layer + NN & 0.75 \\ \hline Interaction-based Conv. Layer: & \\ Window size: \(3\times 3\) & \\ (16 modules listed in Table 4) & \\ Using \(\hat{Y}\) defined in Equation (11) & \\ Interaction-based Conv. Layer + NN & 0.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The performance of a simulation. In this simulation, we defined the underlying model to be (7). The theoretical prediction rate is 75%. A conventional model such as Net-3 and LeNet-5 does not perform well in this example [28,29]. The proposed method that uses the I-score has prediction performance that is close to the theoretical prediction rate (see Section 4.4 for a detailed discussion of AUC values). NN, neural network.