## 3 Feature Selection and Decision
### Feature Selection Using Neighbor Component Analysis (NCA)  
Feature selection involves selecting essential features from the original feature set. This implies that only a few features affect the target BPs. Thus, reducing the dimensions of the feature space while retaining only valid information is essential for BP estimation.  
Weighted feature vectors were extracted from the original feature set using the RNCA algorithm [39]. The NCA method [44] trains a weighted feature vector by minimizing a loss function, in which diagonal adaptation measures the mean deviation leave-one-out regression loss from the training dataset. Thus, a dataset is defined as  
\[D_{w}=\sum_{m=1}^{p}w_{m}^{2}|x_{im}-x_{jm}|, \tag{8}\]  
\begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline  & \multicolumn{2}{c}{**F-test**} & \multicolumn{3}{c}{**RNCA**} & \multicolumn{3}{c}{**F-test**} & \multicolumn{3}{c}{**RNCA**} \\
**Rank** & **SBP** & **DBP** & **SBP** & **DBP** & **Rank** & **SBP** & **DBP** & **SBP** & **DBP** \\ \hline
1 & dppgH & HR & HR & HR & 14 & S2 & S4 & PPGk & MXAP \\
2 & ddppgPH & PAT1 & ST & ST & 15 & PPGk & PIR & S4 & IPA \\
3 & PAT2 & dppgH & DT & ddppgH & 16 & MXAP & MEU & FHR & MIAP \\
4 & ddppgH & ddppgPH & PAT3 & AI & 17 & DT & IPA & MXAP & MEU \\
5 & PAT1 & PAT3 & ddppgW & PAT1 & 18 & MEU & DT & MEU & DT \\
6 & ddppgFH & ST & ddppgFH & PAT3 & 19 & LASI & dppgW & IL & dppgH \\
7 & ST & ddppgH & AI & S2 & 20 & AI & FHR & IPA & ddppgPH \\
8 & PAT3 & LASI & PAT1 & ddppgW & 21 & PIR & ddppgW & dppgW & S4 \\
9 & HR & PAT2 & S3 & ddppgFH & 22 & FHR & MXAP & PAT2 & S1 \\
10 & S3 & S3 & S2 & FHR & 23 & S4 & S1 & LASI & LASI \\
11 & dppgW & ddppgFH & ddppgH & dppgW & 24 & IPA & MIAP & S1 & PIR \\
12 & ddppgW & S2 & dppgH & S3 & 25 & MIAP & AI & PIR & PAT2 \\
13 & S1 & PPGk & ddppgPH & PPGk & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: The high-score ranked features were selected using the F-test and RNCA algorithms for the SBP and DBP estimations.  
Figure 4: The weighted feature sets were obtained using the F-test and RNCA algorithms, where (**a,b**) denote the ranked features acquired using the F-test and (**c,d**) were obtained from the RNCA.  
Here, \(D_{w}\) is the weighted distance and \(w_{m}\) denotes the \(m\)th weighted feature. Thus, the probability \(P(\gamma(x)=x_{j}|\mathcal{T}_{d})\) that point \(x\) is chosen from \(\mathcal{T}_{d}\) as the reference point is:  
\[P(\gamma(x)=x_{j}|\mathcal{T}_{d})=\frac{\mathrm{k}(D_{w}(x_{i}-x_{j}))}{\sum_{ j=1}^{n}\mathrm{k}(D_{w}(x_{i}-x_{j}))} \tag{9}\]  
Here, \((\mathrm{k}(z)=\exp(-z/\sigma))\) denotes the kernel, and the kernel width \(\sigma\) is a parameter that affects the probability that each point is chosen as a reference [44]. Here, we assumed that \(P(\gamma(x)=x_{j}|\mathcal{T}_{d})\propto\mathrm{k}(D_{w}(x_{i},x_{j}))\) and estimated the response to \(\mathrm{x}_{i}\) using the training data set in \(\mathcal{T}_{d}^{-i}\), \((x_{i},y_{i})\). The probability that \(x_{j}\) is chosen as the reference point for \(x_{i}\) is given as  
\[\gamma_{ij=}P(\gamma(x)=x_{j}|\mathcal{T}_{d}^{-i})=\frac{\mathrm{k}(D_{w}(x_ {i}-x_{j}))}{\sum_{j=1,j\
eq i}^{n}\mathrm{k}(D_{w}(x_{i}-x_{j}))} \tag{10}\]  
\[\mathcal{L}_{i}=\mathbf{E}(\mathcal{L}(y_{i},\hat{y}_{i})|\mathcal{T}_{d}^{-i })=\sum_{j=1,j\
eq i}^{n}\gamma_{ij}\mathcal{L}(y_{i},y_{j}) \tag{11}\]  
where \(\mathcal{L}\) is the loss function that gives the difference between \((\hat{y}_{i},y_{i})\). Thus, we applied the regularization parameter \(\lambda\) to minimize the loss function as follows:  
\[\mathcal{F}_{w}=\frac{1}{n}\sum_{i=1}^{n}\mathcal{L}_{i}+\lambda\sum_{m=1}^{p }w_{m}^{2} \tag{12}\]  
Hence, we used the regularization parameter to choose weighted feature vectors from high-dimensional features employing the NCA algorithm as given in (8)-(12), with \(\lambda(=0.015)\)[44].