## 3 Research Method and Materials
### Xtreme Gradient Boost (XGBoost)  
We have adopted the XgBoost algorithm from [PERSON] and [PERSON] [40]. XGBoost is designed for given datasets that have \(n\) examples and \(m\) features, and it is represented by \(\mathcal{D}\{(x_{i},y_{i})\}\) (\(|\mathcal{D}|=n,x_{i}\in\mathbb{R}^{m},y_{i}\in\mathbb{R}\)). By using the \(K\) additive function in Equation (2), it can be used for predicting the output:  
\[\hat{y}_{i}=(x_{i})=\sum_{k=1}^{K}f_{k}(x_{i})\text{,}f_{k}\in\mathcal{F} \tag{2}\]  
where \(\mathcal{F}=\Big{\{}f(x)=\omega_{q(x)}\Big{\}}(q:\mathbb{R}^{m}\to T\), \(\omega\in\mathbb{R}^{T})\) is the spatial regression of trees, \(q\) represents the tree structures, and \(T\) represents the number of leaves on the tree. The total score will be given by \(\omega\) by adding up the corresponding leaves. The regularized objective for learning a set of functions is given by Equations (3) and (4). \[\mathcal{L}()=\sum_{i}l(\hat{y}_{i}-y_{i})+\sum_{k}\Omega(f_{k}) \tag{3}\]  
\[\textit{where }\Omega(\text{f})=\gamma\text{T}+\frac{1}{2}\lambda\|\omega\|^{2} \tag{4}\]  
The given tree representing Equation (3) cannot be trained using Euclidian distances; therefore, [PERSON] and [PERSON] [40] used an additive approach, as shown in Equation (5). \(\hat{y}_{i}^{(t)}\) represents the prediction of the \(i\)-\(th\) instance at the \(t\)-\(th\) iteration. \[\mathcal{L}^{(t)}=\sum_{i=1}^{n}l\Big{(}y_{i}\text{,}\hat{y}_{i}^{(t-1)}+f_{t} (x_{i})\Big{)}+\Omega(f_{t}) \tag{5}\]  
Equation (5) can be improved further with a second-order approximation, as depicted in Equation (6):  
\[\mathcal{L}^{(t)}\simeq\sum_{i=1}^{n}l\Big{(}y_{i}\text{,}\hat{y}_{i}^{(t-1)} \Big{)}+g_{i}f_{t}(x_{i})+\frac{1}{2}h_{i}f_{t}^{2}(x_{i})+\Omega(f_{t}) \tag{6}\]  
Figure 5: Proposed methodology for DCGAN-InceptionV3. where \(g_{i}=\partial_{y_{i}^{(t-1)}}l\Big{(}y_{i},y_{i}^{(t-1)}\Big{)}\) and \(h_{i}=\partial_{y_{i}^{(t-1)}}^{2}l\Big{(}y_{i},y_{i}^{(t-1)}\Big{)}\), and both \(g_{i}\) and \(h_{i}\) are loss functions for the first and second gradient statistics at step \(t\). Equation (6) can be simplified more by removing the constant term; thus, the equation will transform into Equation (7) [40]. \[\widetilde{\mathcal{L}}^{(t)}=\sum_{i=1}^{n}\Big{[}g_{i}f_{t}(x_{i})+\frac{1}{2 }h_{i}f_{t}^{2}(x_{i})\Big{]}+\Omega(f_{t}) \tag{7}\]  
If we define \(I_{j}=\{i|q(x_{i})=j\}\) as an instance that symbolizes leaf \(j\), then Equation (7) can be revised as described in Equations (8) and (9) by expanding \(\Omega\). \[\widetilde{\mathcal{L}}^{(t)}=\sum_{i=1}^{n}\Big{[}g_{i}f_{t}(x_{i})+\frac{1}{ 2}h_{i}f_{t}^{2}(x_{i})\Big{]}+\gamma\mathrm{T}+\frac{1}{2}\lambda\sum_{j=1}^ {T}w_{j}^{2} \tag{8}\]  
\[=\sum_{j=1}^{T}\left[\left(\sum_{i\in I_{j}}g_{i}\right)w_{j}+\frac{1}{2} \left(\sum_{i\in I_{j}}h_{i}+\lambda\right)w_{j}^{2}\right]+\gamma\mathrm{T} \tag{9}\]  
For the static value of \(q(x)\), the optimum weight \(w_{j}^{*}\) of leaf \(j\) is given by Equations (10) and (11). \[w_{j}^{*}(q)=-\frac{\sum_{i\in I_{j}}g_{i}}{\sum_{i\in I_{j}}h_{i}+\lambda} \tag{10}\]  
\[\mathcal{L}()=\sum_{i}l(\hat{y}_{i}-y_{i})+\sum_{k}\Omega(f_{k}) \tag{3}\]0  
Algorithm 1 describes the proposed approach's process flow:  
```
Input: Chest X-ray Image \(l(x,y)\) Step 1: Preprocessing, convert image to grayscale, then apply normalization. \(l:\{\mathbb{X}\subseteq\mathbb{R}^{n}\}\rightarrow\{Min,\ldots,Max\}\) become. \(I_{N}:\{\mathbb{X}\subseteq\mathbb{R}^{n}\}\rightarrow\{newMin,\ldots,newMax\}\).