## 2 Methodology
### Factor Analysis of Anthropometric Parameters  
In order to obtain a concise formula between weight coefficients and anthropometric parameters, we used a factor analysis to select the key anthropometric parameters related to HRTF before formula derivation. There are 27 anthropometric parameters of the subjects in the CIPIC HRTF database (see Table 1), where \(x_{1}\)-\(x_{8}\) are the anthropometric parameters of the head and neck, \(x_{9}\)-\(x_{17}\) are the anthropometric parameters of the torso, and \(d_{1}\)-\(d_{8}\), \(\theta_{1}\), and \(\theta_{2}\) are the anthropometric parameters of the pinna. Factor analysis can be written in a matrix form [32]:  
\[\mathbf{X}=\mu+\mathbf{\alpha}\;F+\varepsilon \tag{12}\]  
where \(\mathbf{X}_{ij}=[x_{1}\;x_{2}\;\ldots\;x_{17}\;d_{1}\;d_{2}\;\ldots\;d_{7}\;\theta_ {1}\;\theta_{2}]\) are the 27 anthropometric parameters of 30 subjects, \(\mu\) is the mean value of \(\mathbf{X}\) across subjects, \(\mathbf{\alpha}\) is the factor loading matrix, \(F\) is the common factor vector, and \(\varepsilon\) is the special factor vector. The steps in factor analysis are as follows:  
Step 1: Perform normalization on \(\mathbf{X}_{ij}\) (\(i=1,\,2,\,\ldots\;,30;j=1,\,2,\,\ldots\;,27\)), eliminating differences in dimensions and the order of magnitude. \[\overline{\mathbf{X}}_{ij,j}=\frac{\mathbf{X}_{i,j}-\overline{x_{j}}}{\sqrt{S_{j}}}\]  
\[\overline{x}_{j}=\frac{1}{30}\sum_{i=1}^{30}X_{i,j} \tag{13}\]  
where \(\overline{x}_{j}\) is the mean value of \(X_{i,j}\), and \(\sqrt{S_{j}}\) is the standard deviation of \(X_{ij}\). Step 2: Estimate the factor load matrix \(\mathbf{\alpha}\) and special variance matrix \(\mathbf{D}\) using maximum likelihood estimation [33]. Step 3: Calculate the variance contribution rate and the cumulative variance contribution rate \(C\). According to the cumulative variance contribution rate, we have selected 16 common factors with a cumulative variance contribution rate of about 90%. \[C=100\%\times\frac{\sum_{n=1}^{16}\alpha^{2}}{16} \tag{14}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable** & **Measurement** & **Type** & **Variable** & **Measurement** & **Type** \\ \hline \(x_{1}\) & head width & head & \(x_{15}\) & seated height & torso \\ \(x_{2}\) & head height & head & \(x_{16}\) & head circumference & torso \\ \(x_{3}\) & head depth & head & \(x_{17}\) & shoulder circumference & torso \\ \(x_{4}\) & pinna offset down & head & \(d_{1}\) & cavum concha height & pinna \\ \(x_{5}\) & pinna offset back & head & \(d_{2}\) & cymba concha height & pinna \\ \(x_{6}\) & neck width & neck & \(d_{3}\) & cavum concha width & pinna \\ \(x_{7}\) & neck height & neck & \(d_{4}\) & fossa height & pinna \\ \(x_{8}\) & neck depth & neck & \(d_{5}\) & pinna height & pinna \\ \(x_{9}\) & torso top width & torso & \(d_{6}\) & pinna width & pinna \\ \(x_{10}\) & torso top height & torso & \(d_{7}\) & intertragal incisure width & pinna \\ \(x_{11}\) & torso top depth & torso & \(d_{8}\) & cavum concha depth & pinna \\ \(x_{12}\) & shoulder width & torso & \(\theta_{1}\) & pinna rotation angle & pinna \\ \(x_{13}\) & head offset forward & torso & \(\theta_{2}\) & pinna flare angle & pinna \\ \(x_{14}\) & height & torso & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of anthropometric parameters in the CIPIC database [7]. Step 4: The maximum variance orthogonal rotation method is used to rotate the factor load matrix \(\mathbf{\alpha}\) to have greater practical meaning [33]. The purpose of the maximum variance rotation method is to differentiate the absolute values of the elements on each column of the factor load matrix \(\mathbf{\alpha}\), that is, the absolute values of a few elements are as large as possible, whereas the other elements as small as possible to zero. Step 5: Finally, the Barlett factor score is estimated by the weighted least squares method [33]. The above factor analysis was conducted through the function Factoran in the MATLAB software.