## 4 Application
### Performance: Proposed Models and AUC Values

Table 10: A summary of the statistics of the design of the proposed network: interaction-based convolutional neural network (ICNN), for Models 1–6. Each model can take one or two interaction-based convolutional layers (i.e., 1 st Conv. or 2 nd Conv. Layer). The model can be designed to directly proceed from the interaction-based convolutional layer to the output layer. For example, Models 1 and 3 proceed directly from the convolutional layer to the output layer, i.e., no hidden layer. variables. In addition to Model 5, it has one hidden layer with 64 units. We fully connect the convolutional layer of 4621 variables with the hidden layer of 64 variables. This results in \(4621\times 64=295,744\) parameters. The hidden layer with 64 units are then fully connected with the output layer, which produces \(64\times 2=128\) parameters. In total, the model has \(295,744+128=295,872\) parameters. This model has the highest AUC value on test set, i.e., \(99.8\%\). The proposed method was also tested on a larger data set [30; 31]. Previous researchers have used portions of this data set [30; 31] and sought to create high-performance CNNs. However, most of these models use millions of parameters and there is no explanation regarding why and what filters (kernels) are used in their CNNs. As a comparison, the proposed method was tested on this multiclass data set as well. The number of sample sizes for training, validating, and test set are summarized in Table 11. While the previous application investigated a data set with binary classes (COVID-19 versus non-COVID-19), this dataset investigates four different classes. These classes are the different variants in lung cancer diseases, which are COVID-19, pneumonia, and tuberculosis. Hence, this data set is a four-class classification task. The response variable for this data set is defined as:  
\[Y=\left\{ \begin{array}{ll}0&\text{if Healthy}\\ 1&\text{if COVID-19}\\ 2&\text{if Pneumonia}\\ 3&\text{if Tuberculosis}\end{array} \right. \tag{21}\]  
Since the target variable \(Y\) takes four values, the computation of the I-score needs to computed class-wise. In other words, \(Y\) is set to one or zero for each class in order to compute the I-score and to generate the proposed convolutional features. A one-hot encode can allow computation in parallel and can reduce computation cost for end users. The data set has a total 994 images with four classes (three types of lung cancer variants and the healthy class). It is a challenging data set in that there is no number of observations required for standard deep learning algorithms. This is another reason why previous scholars have used many convolutional layers in order to extract high-dimensional features in the data [30; 31]. Whereas the previous model delivered relatively good performance, the explainability is somewhat lost throughout the architecture due to many hidden layers. The proposed method was tested on this data set as well and it delivered similar state-of-the-art performance while reducing the number of parameters by almost 99%. With the proposed convolutional features based on the I-score, backward dropping algorithm, and the interaction-based features, the extremely predictive information can be extracted in as little as one layer. In addition, a single hidden layer is recommended to boost performance to the level of its peers; this design has approximately 12,000-13,000 parameters, which is much less than the millions of parameters needed for deep CNNs. The results of this multiclass experiment is summarized in Table 12. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Classes** & **Train** & **Validate** & **Test** \\ \hline Healthy & 437 & 44 & 52 \\ Tuberculosis & 422 & 41 & 52 \\ Pneumonia & 88 & 9 & 1 \\ COVID-19 & 88 & 9 & 11 \\ Total & 994 & 99 & 121 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Multiclass prediction data summary of the number of training set, validating set, and testing set samples in the multiclass X-ray image classification.