## 3 Methodology
### Partial Least Square Regression for HRTF Synthesis  
Partial least square regression is a statistical method that is used to search the basic functional relationship between two matrices [23]. It is often used in mapping the linear relationship from multiple independent variables to multiple dependent variables.  
Assuming that the optimized anthropometric features of \(S\) subjects and the logarithmic magnitudes of HRTFs at \(M\) directions are presented with the matrices:  
\[X_{F\times S}=\left[\begin{array}{cccc}x_{11}&x_{12}&\cdots&x_{1S}\\ x_{21}&x_{22}&\cdots&x_{2S}\\ \vdots&\vdots&\ddots&\vdots\\ x_{F1}&x_{F2}&\cdots&x_{FS}\\ \end{array}\right] \tag{9}\] \[\left[\mathbf{H}_{\log,f}(\theta_{1},S_{1})\right.\left. \mathbf{H}_{\log,f}(\theta_{1},S_{2})\right.\left.\cdots&\mathbf{H}_{\log,f} (\theta_{1},S_{S})\\ \mathbf{H}_{\log,f}(\theta_{2},S_{1})\right.\left.\mathbf{H}_{\log,f}(\theta _{2},S_{2})\right.\left.\cdots&\mathbf{H}_{\log,f}(\theta_{2},S_{S})\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{H}_{\log,f}(\theta_{M},S_{1})\right.\left.\mathbf{H}_{\log,f}(\theta _{M},S_{2})\right.\left.\cdots&\mathbf{H}_{\log,f}(\theta_{M},S_{S})\\ \end{array}\right]\]  
where \(F\), \(S\), \(M\) and \(N\) represent the number of anthropometric features used in the model, training subjects, training azimuths and the frequency bins of HRTF, respectively.  
The first step of PLRR is to normalize the dependent and independent matrices. The normalization process is conducted as the equation below:  
\[a_{ij}^{*}=\frac{a_{ij}-m_{ij}}{M_{ij}-m_{ij}} \tag{10}\]  
where \(a_{ij}^{*}\) stands for the normalized value of \(a_{ij}\), \(M_{ij}\) and \(m_{ij}\) stand for the maximum and minimum value of \(a_{ij}\), respectively. The normalized matrices are \(\mathbf{X}_{0}\) and \(\mathbf{H}_{0}\).  
The second step is to construct the internal and external relationships between dependent variables and independent variables, and implement the regression of \(\mathbf{X}_{0}\) and \(\mathbf{H}_{0}\) on the first component:  
\[\mathbf{X}_{0}=\mathbf{t}_{1}\mathbf{p}_{1}^{T}+\mathbf{X}_{1} \tag{11}\]  
\[\mathbf{H}_{0}=\mathbf{t}_{1}\mathbf{r}_{1}^{T}+\mathbf{H}_{1} \tag{12}\]  
Figure 6: The anthropometric features used in the presented integrated method. x\({}_{14}\) is the seated height, which is not marked in the figure.  
where \(\mathbf{t}_{1}\) represents the first component extracted from \(\mathbf{X}_{0}\), \(\mathbf{p}_{1}\) and \(\mathbf{r}_{1}\) represent the regression coefficients, \(\mathbf{H}_{1}\) and \(\mathbf{X}_{1}\) represent the residual of \(\mathbf{X}_{0}\) and \(\mathbf{H}_{0}\), respectively.  
Next, \(\mathbf{X}_{0}\) and \(\mathbf{H}_{0}\) are replaced with \(\mathbf{X}_{1}\) and \(\mathbf{H}_{1}\), and the second step is repeated for \(l\) times until the accuracy of regression equation met the requirement, where \(l\) is determined by the cross validation seeing [22]. Then, the estimation of \(\mathbf{H}_{0}\) can be denoted as:  
\[\mathbf{\hat{H}}_{0}=\sum_{i=1}^{l}\mathbf{t}_{i}\mathbf{r}_{i}^{T} \tag{13}\]  
Finally, according to the inverse process of normalization, the regression equation can be restored. Figure 7 shows an example data of measured and synthesized HRTF on the left ear of a KEMAR dummy head at \(\theta=0^{\circ}\) on the horizontal plane.