# Individualized Interaural Feature Learning and Personalized Binaural Localization Model +
## 4 Feature Dependency Analysis and Assembled Data Partition Model
### Data Partition and Tree-Structured Model  
The dependency analysis described in Section 2 justifies the existence of elevation-related features at full-band ILD and high-frequency IPD. To use these results and further exploit the feature characteristics, the mapping model should satisfy two essential requirements. First, the model should be able to handle both linear and non-linear associations. Second, an adaptive feature selection should be embedded in the model training progress. Therefore, to fulfil these two requirements, a recursive tree-structured data partition technique--the so-called decision tree method--is introduced. A tree is a hierarchical structure in which each internal node contains a specific feature condition. At each node, the training data are split into two sub-spaces by the splitting feature that maximizes the MI between the selected feature and the spatial labels from sub-spaces. Notably, the MI here is also known as information gain in machine learning, which is still denoted by \(\mathcal{IG}\) in latter content. Finally, each node at the terminal of the tree is defined as a leaf node, which represents a  
which can also be interpreted as subtraction of entropies by:  
\[\mathcal{IG}(\mathrm{y},v_{m,k})=H(\mathrm{y})-H(\mathrm{y}|v_{m,k}). \tag{13}\]  
It is more directly perceived through (13) that the \(\mathcal{IG}\) is evaluating the decreasing uncertainty of \(\mathrm{y}\) after splitting by \(v_{m,\mu}\). The probability components \(P_{m}(v_{m,\mu},\mathrm{y}_{d})\), \(P_{m}(v_{m,\mu})\) and \(P_{m}(\mathrm{y}_{d})\) are estimated by histogram-based probability estimation [28] as:  
\[P_{m}(v_{m,\mu},\mathrm{y}_{d}) =\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}z(v_{m,\mu}^{(n)}\leq v_{m,\mu},\mathrm{y}=\mathrm{y}_{d})\] \[P_{m}(v_{m,\mu}) =\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}z(v_{m,\mu}^{(n)}\leq v_{m,\mu}) \tag{14}\] \[P_{m}(\mathrm{y}_{d}) =\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}z(y_{m,d}^{(n)}=\mathrm{y}_{d})\]  
where the indicator function \(z(\cdot)\) is defined as same as (9). Depending on (12) and (14), one may split the dataset by selecting the feature index \(\mu\) and its splitting value \(v_{m,\mu}\), which maximizes the value of information gain as:  
\[\tilde{v}_{m,\mu}=\underset{v_{m,\mu}\in\mathcal{V}_{m}}{\arg\max}\mathcal{IG }(\mathrm{y},v_{m,k}) \tag{15}\]  
where \(\tilde{v}_{m,\mu}\) indicates the optimized splitting value with its feature index \(\mu\) at node \(m\). By repeating this splitting operation until some certain criteria is reached (e.g., the maximum number of nodes \(M\)), the original dataset \(\{\mathbf{v}^{(n)},\mathbf{y}^{(n)}\}_{n=1}^{N}\) is partitioned into multiple subsets denoted as \(\{\mathbf{v}_{\lambda}^{(n)},\mathbf{y}_{\lambda}^{(n)}\}_{n=1}^{N_{\lambda}}\) with subset index \(\lambda\). The whole training progress can also be understood as a recursive clustering process, whereby the training data with common feature characteristics described by \(v_{k}^{m}\) are clustered into the same subset \(\lambda\), while the uncertainty of spatial label \(\alpha\) is minimized, so that, in such a subset, the training data only share a few \(\alpha\). Further, in this recursive approach, the splitting feature index \(k\) and its corresponding value \(v_{k}^{m}\) are adaptively re-selected via (15) at each node, so the most spatial dependent feature characteristics are exploited to the maximum.