## 3 Proposed Method
### Simulation with Artificial Examples
#### 3.5.1 Artificial Example I: Variable Investigation
\right.\]  
where \(j=\{1,2,3,4\}\), which corresponds to the \(2^{2}=4\) partitions generated by \(\{X_{1},X_{2}\}\). This provides the first predictor to allow us to build classifier and to make predictions. The experimental results from the first artificial example are listed in Tables 3-5. Suppose a statistician knows the model. The theoretical prediction rate is 75% (which is derived using Formulas (8) and (9)). In reality, we suppose that a statistician has no knowledge of the data set. In this case, they can directly proceed using a neural network or even CNN to make predictions. However, without the correct model specification, the performance for these models is subpar. We can see that the neural network and CNN both performed 50% on he test set. This performance is rather like random guessing. Alternatively, this statistician can perform the experiment using the proposed methods. First, we can generate an interaction-based convolutional layer. Instead of using many pretrained filters for feature mapping, the proposed method adopts the I-score and the backward dropping algorithm to construct \(X_{\{X_{1},X_{2}\}}^{\dagger}\), \(X_{\{X_{k},X_{3},X_{21}\}}^{\dagger}\), and so on. The proposed methodology also has an exact I-score that is associated to each variable module, which allows us to rank feature importance. With the I-score, the first variable module \(X_{\{X_{1},X_{2}\}}^{\dagger}\) is much more influential than the other variable modules. We proceed using a window size of \(2\times 2\). In a data set with \(6^{2}=36\) variables, this produces \((6-2+1)^{2}=25\) variable modules. Using these 25 variable modules, we are able to build a classifier and achieve a performance of 76% AUC (see Section 4.4 for a detailed discussion of AUC values) on the test set, much higher than the previous 50%. We can also use a window size of \(3\times 3\). This produces \((6-3+1)^{2}=16\) variable modules. These 16 modules allow us to build another classifier using a neural network and delivers a 76% AUC on the test set as well. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**New Mod.** & **Variables** & **I-Score** & **AUC** & **New Mod.** & **Variables** & **I-Score** & **AUC** \\ \hline \(X_{1}^{\dagger}\)* & X1, X2 & 638.17 & 0.75 & \(X_{14}^{\dagger}\) & X28 & 1.3729 & 0.50 \\ \(X_{2}^{\dagger}\) & X7 & 1.2162 & 0.50 & \(X_{15}^{\dagger}\) & X28 & 1.3729 & 0.50 \\ \(X_{3}^{\dagger}\) & X13, X20 & 2.3597 & 0.51 & \(X_{16}^{\dagger}\) & X11 & 0.2347 & 0.50 \\ \(X_{4}^{\dagger}\) & X19, X20, X26 & 0.7218 & 0.50 & \(X_{17}^{\dagger}\) & X11 & 0.2347 & 0.50 \\ \(X_{5}^{\dagger}\) & X26, X31 & 2.6751 & 0.50 & \(X_{18}^{\dagger}\) & X16, X22 & 0.0777 & 0.51 \\ \(X_{6}^{\dagger}\) & X8, X9 & 0.5067 & 0.49 & \(X_{19}^{\dagger}\) & X28 & 1.3729 & 0.51 \\ \(X_{2}^{\dagger}\) & X8, X9 & 0.5067 & 0.50 & \(X_{20}^{\dagger}\) & X28 & 1.3729 & 0.51 \\ \(X_{8}^{\dagger}\) & X15, X21 & 1.8013 & 0.50 & \(X_{21}^{\dagger}\) & X6, X12 & 0.4378 & 0.49 \\ \(X_{20}^{\dagger}\) & X20, X21, X26, X27 & 0.7554 & 0.50 & \(X_{22}^{\dagger}\) & X11, X12 & 0.6184 & 0.51 \\ \(X_{10}^{\dagger}\) & X27, X32 & 1.017 & 0.50 & \(X_{23}^{\dagger}\) & X18, X24 & 1.3814 & 0.51 \\ \(X_{11}^{\dagger}\) & X9, X10 & 0.6894 & 0.50 & \(X_{24}^{\dagger}\) & X23, X24, X29 & 0.8788 & 0.51 \\ \(X_{12}^{\dagger}\) & X9, X10, X15 & 0.9346 & 0.51 & \(X_{25}^{\dagger}\) & X30, X35 & 1.2105 & 0.51 \\ \(X_{13}^{\dagger}\) & X15, X16, X21, X22 & 1.0933 & 0.50 & & & & \\ \hline \end{tabular}
\end{table} 
Table 3: Interaction-based convolutional layer. Foran artificial data set with \(6^{2}=36\) variables and a window size of \(2\times 2\), we pass the window from the top left corner down to the bottom right corner. For each particular location, we have 4 variables to run the BDA. Before each observation had 36 features that could be sized \(6\times 6\). Afterward, each observation has 25 new features that have the shape of \(5\times 5\). In other words, we create \(X_{b}^{\dagger}\), where \(b=\{1,2,\dots,25\}\).