## 3 Methodology
### Multiple Linear Regression for Anthropometric Feature Selection  
In this research, multiple linear regression was used for selecting the anthropometric features closely relating to the HRTFs. Suppose the relationship between principle components and the corresponding anthropometric features in the spatial direction \(\theta\) is:  
\[\mathbf{w}_{\theta}=\mathbf{X}\mathbf{B}_{\theta}+\mathbf{E}_{\theta} \tag{5}\]  
where \(\mathbf{w}_{\theta}\) represents the weight vector of HRTF logarithmic magnitude at direction \(\theta\), \(\mathbf{B}_{\theta}\) represents the regression coefficients matrix, \(\mathbf{X}\) denotes the anthropometric feature matrix, \(\mathbf{E}_{\theta}\) represents the estimation error matrix, respectively. The regression coefficients matrix \(\mathbf{B}_{\theta}\) can be calculated through the anthropometric features:  
\[\mathbf{B}_{\theta}=(\mathbf{X}^{T}\mathbf{X})^{-1}\mathbf{X}^{T}\mathbf{w}_{\theta} \tag{6}\]  
Figure 5: Result of PCA on a left ear’s HRTF logarithmic magnitudes. \(T\) represents the transpose of a matrix. There were a number of anthropometric features in the database. For the reason that there were different correlations between the features and HRTFs, it was unnecessary to introduce all of them into the MLR models. By using unnecessary features, some useful information might be concealed, which led to a worse regression model. Meanwhile, getting rid of the unnecessary features alleviated the complexity of the system, which reduced the workload of the individualization procedure. In order to select the anthropometric features used in the regression model, correlation analysis was first conducted on different features. For the two features with large linear correlation coefficient, the one which had more significant influence was reserved. Then, correlation analysis was applied on the reserved features and HRTF logarithmic magnitudes to delete those features with smaller correlation coefficients. Next, the selected anthropometric features were introduced to the MLR model, in which \(F\)-test and backward selection at significant level \(\alpha=0.05\) would be used to delete those insignificant features. Considering the linear regression model:  
\[\mathbf{Y}=\mathbf{X}\beta+\epsilon \tag{7}\]  
where \(\mathbf{X}\) is an n by p full rank matrix of known constant, \(\mathbf{Y}\) is an n-vector of response, \(\beta\) is a p-vector of unknown parameter, and \(\epsilon\) is an \(n\) by 1 unobservable error with a normal distribution \(N(0,\sigma^{2}I_{n})\). Assume that a hypothesis in model (7) is given by \(H_{0}\): \(\mathbf{A}\beta=\mathbf{c}\), where A is a known \(q\) by \(p\) matrix with rank \(q\) and \(\mathbf{c}\) is a \(q\) by 1 vector. The usual test for hypothesis \(H_{0}\) is \(F\)-test, which was equivalent to the likelihood ratio test. The test statistic F-statistic was given by:  
\[F=\frac{1}{q(\hat{\sigma})^{2}}(\mathbf{c}-\mathbf{A}\hat{\beta})^{T}[ \mathbf{A}(\mathbf{X}\mathbf{X}^{T})^{-1}\mathbf{A}^{T}]^{-1}(\mathbf{c}- \mathbf{A}\hat{\beta}) \tag{8}\]  
where \(\hat{\beta}=(X^{T}X)^{-1}X^{T}Y\) is a least square estimation of \(\beta\), and \(\sigma^{2}=\mathbf{Y}^{T}(\mathbf{I}_{n}-\mathbf{P}_{x})\mathbf{Y}/(n-p)\) is an unbiased estimator of \(\sigma^{2}\) in model (7), where \(\mathbf{P}_{x}=\mathbf{X}(\mathbf{X}^{T}\mathbf{X})^{-1}\mathbf{X}^{T}\). When \(H_{0}\) holds the \(F\)-statistic Equation (8) distributes as an \(F\) distribution with degrees of freedom \(q\) and (\(n\)-\(p\)). After the F-statistic has been calculated, the backward selection is used to select the significant features. First, all the features retained after correlation analysis were put into the regression model, then \(F\)-statistic of each feature was calculated and compared to the one at the significant level \(\alpha=0.05\). After deleting the most insignificant feature, the rest ones were put into the model again and the insignificant one among them was deleted.