# Blood Pressure Estimation by Photoplethysmogram Decomposition into Hyperbolic Secant Waves
## 2 Methods
### The Verification of Our Proposed Method  
In this study, the two estimation methods based on each function-based PDA or Gaussian function-based PDA were subjected to two validations: the first was the accuracy of the model for training (regression) with known data, and the second was the accuracy of the model to predict blood pressure (\"unknown\" data not used for training) with the regression coefficients obtained from the training data.  
In addition, two analyses were performed for each validation. The first wanaan analysis using a within-subject model, which did not consider individual differences. The  
Figure 4: Processed pulse wave.  
Figure 5: Box plots with 60 systolic blood pressure (sbp) for each subject. Each sbp point is the average of an 8-s window.  
Figure 3: Original pulse wave.  
second was an analysis using a between-subjects model, that is, the regression coefficients were obtained using multiple subjects.  
First, multiple regression analysis was performed using all subjects' data without data division. The goodness of fit of the multiple regression analysis was quantified by calculating the multiple correlation coefficients, which determined the extent to which the features achieved a description of blood pressure for all subjects' data. In this test, two analyses were performed, one for each subject data (within-subject test) and the other for all subject data (between-subjects test). We adopted first-order regression models.  
Second, to quantify the estimability for the test data, the Leave One Out method was applied to the dataset, and the correlation coefficient between the ground truth and the estimate was calculated. Similar to the first case, here, two analyses were performed: one for each subject's data and the other for all subject's data. The Leave One Out method was used to split the segment data into training data and test data. These data were divided into 59:1 segments by the Leave One Out method for the within-subject model, and into 599:1 segments for the between-subjects model. The Leave One Out method was conducted with the Python programming language library Scikit-learn, which is a machine learning library.  
Regarding the selected features, we used the feature set that provided the best performance for each subject's data. This was because the optimal feature set for each subject could have been different because of individual differences. For verification with the training data, the feature set (function parameters) that provided the best performance was used to determine multiple correlation coefficients. As a result, all features provided the best performance on all subjects' data. For verification with the test data, the feature set that provided the best performance was used to determine the multiple correlation coefficients. As a result, the features set shown in Table 1 provided the best performance.  
Statistical \(p\)-values were calculated using t-tests for comparison between two quantities with standard deviations. For the determination of the \(p\)-value of the slope of the linear regressions of the ground truth vs. the estimated value of systolic blood pressure, we considered that the null hypothesis was that the slope was zero (no correlation).