## 3 Proposed Method
### Simulation with Artificial Examples
#### 3.5.1 Artificial Example I: Variable Investigation  
In the first artificial example, we demonstrate the procedure followed to construct the interaction-based convolutional layer and the engineer interaction-based features. Let us consider the following experiment. We create an artificial data set with 500 data points in he training set and 10,000 data points in the testing set. These observations are randomly drawn from Bernoulli(1/2) independently. In other words, we independently generate \(X_{i}\sim\text{Bernoulli}(1/2)\), where \(i=\{1,2,3,\ldots,36\}\). We define the underlying model to be the following (here known as Model (7)),  
\[Y=\left\{ \begin{array}{ll}X_{1}+X_{2}&(\text{mod 2})\text{ with prob. }0.5\\ X_{3}+X_{4}+X_{5}&(\text{mod 2})\text{ with prob. }0.5\end{array} \right. \tag{7}\]  
Model (7) is a two-module example. The first module is a two-way interaction in modulo 2. The second module is a three-way interaction in modulo 2. The response variable \(Y\) is defined as 50% first module and 50% second module. In this setup, the individual explanatory variable does not have any predictive power on the response variable \(Y\). **Scenario I. ** Assume the statistician knows the model in this simulation. This means they are fully aware of \(S_{1}=\{X_{1},X_{2}\}\) as an important variable set and \(S_{2}=\{X_{3},X_{4},X_{5}\}\) as the other. In other words, they can use the first module as a predictor to make predictions on response variable \(Y\). They are able to compute the theoretical prediction rate of the first variable set as 75%. This is because the response variable is defined as the first variable module \(S_{1}=\{X_{1},X_{2}\}\) exactly 50% of the times, so \(S_{1}\) is able to guess \(Y\) correctly at least 50% of the time. The other 50% of the time, the response variable is defined as the second variable module \(S_{2}=\{X_{3},X_{4},X_{5}\}\). Since there is no marginal signal, the performance is exactly like random guessing, so the rest of the occurrences are only correct 50% of the time. In other words, assuming training sample size has \(n\) data points, we can write the following:  
\[ \begin{array}{rcl}\theta_{c}(S_{1})&=&\theta_{c}(\{X_{1},X_{2}\})\\ &=&\frac{1}{n}\sum_{i}^{n}\mathds{1}(\hat{Y}=Y)\\ &=&\frac{1}{n}\sum_{i}^{n}\mathds{1}(\underbrace{(X_{1}+X_{2})}_{(\text{mod 2 })}=Y)\\ &=&50\%+50\%\cdot 50\%\\ &=&75\%\end{array}  \tag{8}\]  
This result can be extended to the other variable module as well. If this statistician uses the second variable module \(S_{2}=\{X_{3},X_{4},X_{5}\}\), then a similar calculation can be carried out in the following. \[ \begin{array}{rcl}\theta_{c}(S_{2})&=&\theta_{c}(\{X_{3},X_{4},X_{5}\})\\ &=&\frac{1}{n}\sum_{i}^{n}\mathds{1}(\hat{Y}=Y)\\ &=&\frac{1}{n}\sum_{i}^{n}\mathds{1}(\underbrace{(X_{3}+X_{4}+X_{5})}_{(\text {mod 2})}=Y)\\ &=&50\%+50\%\cdot 50\%\\ &=&75\%\end{array}  \tag{9}\]  
The theoretical prediction rate for Model (7) is the maximum percentage accuracy delivered by \(S_{1}\) and \(S_{2}\). In this case, we have \(\max(\theta_{c}(S_{1}),\theta_{c}(S_{2}))=0.75\). **Scenario II. ** In practice, it is often the case that we do not exactly observe the underlying model in a given data set. The recommendation is to use the I-score to select the important local information and then create an interaction-based feature based on the selected variable. The diagram for this action is depicted in Figure 4.  
The classical procedure of a convolutional neural network starts with many predefined filters (or kernels) from a previous data set. The size of this filter can be \(2\times 2\), \(3\times 3\), or any other higher dimension. The procedure starts with the first row and the first column. Then, the process passes the filter over and across rows and columns in the image. The proposed methodology shares similar characteristics. However, instead of using a pretrained filter, we apply the BDA in this local area. **Constructing the Proposed Convolutional Layer. ** In the proposed artificial example, we have a data set with 36 variables.