# A Boundary Zone Method for the Generation of Multivariate Representative Humanoids
## 5 Discussion  
The present study developed the BZM, which can resolve the limitations (loss of anthropometric variability, lack of body size diversity, and estimation error of an AD) of the existing multivariate RH generation methods (SM, RM, and CM). The BZM uses normalized squared distances of anthropometric cases that follow a \(\chi^{2}\)-distribution--not data reduction techniques such as FA and PCA, which cause the loss of anthropometric variability in RH generation [20]. Next, the BZM uses a statistical clustering method to cluster anthropometric cases in the BZ so that the occurrence of missing zones between RHs can be avoided and the body size diversity of the target population can be better represented. Lastly, the BZM selects a real anthropometric case, not a case of which individual body dimensions are estimated, for an RH of each cluster so that an error due to estimation of ADs can be prevented. The BZM is based on the assumption that ADs follow a multivariate normal distribution to form a BZ using normalized square distances of anthropometric cases. ADs are commonly known as normal [2,7], but significant normality violations exist in some ADs, such as chest breadth and forearm-hand length [28]. The Kolmogorov-Smirnov test was conducted on the 10 ADs of the U.S. Army data considered in the present study and identified that three ADs in males and four ADs in females failed to satisfy the normal distribution assumption at \(\alpha=0.01\). In the present study, non-normal ADs were transformed by the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Anthropometric Dimension** & **Factor 1** & **Factor 2** & **Factor 3** \\ \hline \hline \multicolumn{1}{c}{Political height} & \(\mathbf{0.96}\) & \(0.09\) & \(-0.07\) \\ \multicolumn{1}{c}{Knee height} & \(\mathbf{0.96}\) & \(-0.14\) & \(-0.06\) \\ \multicolumn{1}{c}{Foot length} & \(\mathbf{0.89}\) & \(-0.08\) & \(-0.21\) \\ \multicolumn{1}{c}{Buttock–knee length} & \(\mathbf{0.86}\) & \(-0.41\) & \(0.16\) \\ \multicolumn{1}{c}{Buttock–political length} & \(\mathbf{0.84}\) & \(-0.34\) & \(0.28\) \\ \multicolumn{1}{c}{Forearm-to-forearm breadth} & \(\mathbf{0.55}\) & \(-0.43\) & \(-0.53\) \\ \multicolumn{1}{c}{Hip breadth} & \(-0.10\) & \(\mathbf{-0.88}\) & \(0.08\) \\ \multicolumn{1}{c}{Abdominal extension depth} & \(0.24\) & \(\mathbf{-0.79}\) & \(-0.25\) \\ \multicolumn{1}{c}{Thigh clearance} & \(0.39\) & \(\mathbf{-0.70}\) & \(-0.26\) \\ \multicolumn{1}{c}{Elbow rest height} & \(-0.09\) & \(-0.11\) & \(\mathbf{-0.86}\) \\ \hline \hline \end{tabular} Note. Absolute values of factor loadings \(>\)0.5 are in boldface.  
\end{table} 
Table 3: Factor loadings of anthropometric dimensions. Figure 7: Distribution of the representative humanoids generated by the multivariate methods (illustrated). (**a**) Buttock–knee length vs. hip breadth, with a different factor loading pattern; (**b**) abdominal extension depth vs. thigh clearance, with a similar factor loading pattern. Box-Cox transformation method before establishing a BZ. Since Box-Cox transformation was applied only to identify the RHs, its reverse transformation was not necessary. The BZM requires a technical discretion on the level of tolerance (e.g., \(\delta\) = \(\pm\)1%) and a large amount (>2000) of anthropometric data to construct a proper BZ. If a tolerance level is too small, the number of anthropometric cases within the BZ can become too small to generate RHs in the subsequent steps. Conversely, if a tolerance level is too large, anthropometric cases far apart from the designated \(\kappa\) can be considered as candidates of RHs. Therefore, an appropriate level of tolerance needs to be determined by considering various technical aspects such as the size of the anthropometric database--the larger the size of the anthropometric database, the smaller the tolerance level. The outlier analysis showed that only the BZM did not generate outliers, while the existing methods generated RHs with improper values of ADs beyond their corresponding body size ranges of the target population.