## 3 Case Study and Results
### Results of the Bayesian Search  
Table 2 shows the Bayesian search space for the XGBoost and RF. There are numerous possible combinations for each method. More specifically, there exist around 380 combinations for the RF search space and around 1140 combinations (continues search space can be counting as a step size of 0.01) for the XGBoost search space that need to be explored. It is impossible to use an exhaustive search simply because of the enormous computation burden. Moreover, there are more than 100 body features in the 3D BM dataset and it is computationally infeasible to find out the best combination of the optimal hyperparameters in their search space for the ML methods. However, the Bayesian search can overcome the issue of computational infeasibility and obtain near-optimal combinations of the hyperparameters. In this study, the Hyperopt (i.e., Distributed Asynchronous Hyper-parameter Optimization) package [36] is utilized as a Python library to implement Bayesian search in the proposed framework. The maximum evaluation times is 12 for the search space of both the XGBoost and the RF. A detailed analysis of the feature importance will be provided in the next subsection after the near-optimal combinations of hyperparameters are procured. Figure 6 shows the MSE results after the Bayesian search for the XGBoost and the RF with all features including a total of 117 body features. Only one feature is considered to be predicted--the output of the ML model--while all of the other body features are taken as input to the ML methods. For instance, the blue and yellow bars labeled \"Body Height (in)\" in Figure 6 represents that the body height is the only feature to be predicted while all of the other body features (i.e., 117 - 1 16 features) except for the \"body height\" are taken as input to the ML methods. The blue bar represents the MSE result from XGBoost and the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**ML Method** & **Search Space Parameter** & **Search Type** & **Search Space Range** \\ \hline \multirow{8}{*}{XGBoost} & gamma & continuous & [5,0, 11.0] \\  & learning rate & continuous & [0.07, 0.6] \\ \cline{1-1}  & n estimators & discrete & 50, 100, 150 \\ \cline{1-1}  & regulation alpha & discrete & \(1\times 10^{-5}\), \(1\times 10^{-2}\), \\ \cline{1-1}  & regulation lambda & discrete & \(1\times 10^{-5}\), \(1\times 10^{-2}\), \\ \cline{1-1}  & min child weight & discrete & 0.45 \\ \cline{1-1}  & subsample & discrete & \(1.5\), \(6\), \(10\) \\ \cline{1-1}  & max depth & discrete & \(0.6\), \(0.95\) \\ \hline \multirow{4}{*}{Random Forest} & n estimators & discrete & 100, 200, 500 \\ \cline{1-1}  & max depth & discrete & \(4\), \(5\), \(6\),..., \(11\) \\ \cline{1-1}  & min samples & discrete & \(2\), \(3\), \(4\), \(5\) \\ \cline{1-1}  & min samples leaf & discrete & \(1\), \(3\), \(5\), \(7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hyperparameter of ML models. Figure 5: Distributions of age and gender groups of the dataset. orange bar represents the MSE result from RF as shown in Figure 6. It is seen that some features, e.g., \"body height\", \"weight\", and \"waist to hip ratio\", have comparatively large values; nevertheless, most of other features show comparatively smaller values, i.e., \"waist to buttock height left\" and \"waist to buttock height right\", etc. According to Equation (4), the MSE increases as the difference between the actual value and the predicted value increases. In addition, MSE is scale-dependent which leads to the result having a large value variety among different body features. However, MSE has the ability to obviously display the bias of how the ML model fits the data. Figure 6: MSE results of XGBoost and RF with Optimal hyper-parameter via Bayesian search. It is interesting to compare the performances between the RF and the XGBoost. As seen in Figure 6, the RF has better performance than the XGBoost does in most cases in terms of a smaller MSE. However, there are two exceptions, i.e., the weight and the waist to hip ratio.