## 2 Materials and Methods
### Ankle-Brachial Index (ABI) Prediction by Means of Machine Learning (General Linear Model, GLM)  
Machine learning was utilized to predict ABI relying on features extracted from average brachial and tibial PPG pulses. Specifically, a GLM was trained on the ABI estimated by the VE (VE-ABI) through a supervised learning procedure. The GLM explains a dependent variable (e.g., ABI) as a weighted sum of one or more predictor variables (e.g., PPG features) plus an unknown error term, in order to evaluate the extent of the contribution of the predictors to the variability observed in the dependent variable. The weights (\(\beta\)s) of the model can be trained on the experimental data and, after training on a normalized dataset, they are informative of the strength of the relation between the predictors and the predicted variable. The GLM can be expressed as [28]:  
\[\text{Y}=\text{X}\beta+\varepsilon \tag{2}\]  
where  
\(\bullet\)\(\text{Y}=\text{n}\times 1\) column vector representing the dependent variable (e.g., VE-ABI)  
\(\bullet\)\(\text{X}=\text{n}\times\text{p}\) design matrix where each column represents an independent variable of length n (e.g., PPG features);  
\(\bullet\)\(\beta=\text{p}\times 1\) column vector of weights of each predictor, indicating the strength of the association with Y;  
\(\bullet\)\(\varepsilon=\text{n}\times 1\) column vector of the residual error  
The optimization of the parameters \(\beta\)s (training), generally based on a least square approach and performed on normalized (z-scored) variables, provides information regarding the amount of variance of the dependent variable that could be explained by each independent variable. A total of 7 independent variables (features of the PPG signals), supposedly sensitive to cardiovascular status [41], were chosen for the analysis, namely:  
Figure 3: Preprocessing steps for ECG and PPG signals and an example of pulse average ECG and PPG and their standard errors (shaded areas), showing the small variability of the signals. * Maximum amplitude of the PPG ODs evaluated at brachial and tibial arteries (2 features, labelled MaxArm and MaxAnkle);
* Diastole to systole slope of the PPG signal at brachial and tibial arteries (2 features, labelled SlopeArm and SlopeAnkle) evaluated as: \[\textit{Slope}=\frac{\textit{Systolic peak}-\textit{Diastolic foot}}{\textit{time Systolic peak}-\textit{time Diastolic foot}}\] (3)
* Time delay of the diastolic foot of PPG with respect to ECG R-peak at brachial and tibial arteries (2 features, labelled TDArm and TDAnkle);
* Ratio of an estimate of the systolic blood pressure at the ankle and at the brachial artery derived from PPG signals according to [42] (1 feature, labelled Ankle-Arm), evaluated as: \[\textit{Ankle}-\textit{Arm}=\frac{184.3-1.329\cdot\textit{HeartRate}_{ppm}+0.0848 \ast TD_{\textit{m ankle}}}{184.3-1.329\cdot\textit{HeartRate}_{ppm}+0.0848 \ast TD_{\textit{arm}}}\] (4)  
Importantly, not only PPG features intuitively directly associated to pressure features employed in ABI computations were taken into account in the regression problem (such as the maximum amplitudes), but also other features that could be sensitive to the cardiovascular status (such as the heartrate, PPG slope and timing). The features were normalized (z-scored) before GLM computation. Left and right metrics were considered as independent samples in the GLM regression problem, hence resulting in a sample size of 146 (73 participants \(\times\) 2 body sites, left and right). Because of the multivariate (7 regressors) GLM approach, in-sample performance of the procedure did not reliably estimate the out-of-sample performance. The generalization capabilities of the procedure were thus assessed through cross-validation. Specifically, a leave-one-subject-out cross-validation was performed [43].