## 2 Methods
### Multi-Dimension Regression Model  
In this study, we used the linear multi-dimension regression model to estimate the blood pressure, where the model explores the correlation between the independent variable (\(x\)) and the dependent variable (\(y\)), and establishes a regression model to estimate the variables (\(\hat{y}\)). Let y be the dependent variable, whilst \(x_{1}\), \(x_{2}\),... \(x_{k}\) are the independent variables. There is a linear relationship between the  
\begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Formula** \\ \hline Ejection Time Ratio (ETR) & ETR = \(\frac{T4+T5}{T1+T2+T3}\) \\ Beat Duty (BD) & BD = \(T1+T2+T3\) \\ Heart Rate (HR) & HR = \(\frac{\text{BD}}{\text{TPG}_{\text{v}}}\) \\ Volume change during systolic cycle (Z0 AS) & Z\({}_{\text{QAS}}\) = \(\frac{\text{TPG}_{\text{v}}}{\text{TPG}_{\text{v}}}\) \\ Volume change during diastolic cycle (Z0 DS) & Z\({}_{\text{QDS}}\) = \(\frac{\text{TPG}_{\text{v}}}{\text{TPG}_{\text{v}}}\) \\ Volume during systolic cycle (Z0 AA) & Area during T1 in the pulse wave. \\ Volume during diastolic cycle (Z0 DA2) & Area during T3 in the pulse wave. \\ Maximum volume change during systolic cycle (DPPGp) & Peak between T4 and T5 in the differential wave of pulse. \\ Minimum volume change during diastolic cycle (DPPGv) & Valley at the end of T5 in the differential wave of pulse. \\ Pulse transit time (PTT) & Time interval from the systolic ending of the left \\  & ventricular to the specific position of the peripheral artery. \\ \hline \end{tabular}
\end{table}
Table 1: Parameter definition of the pulse wave.  
Figure 1: Red line represents the pulse wave, blue line represents the forward wave, light blue line represents the reflected wave.  
Figure 2: (a) The pulse wave, (b) The differential wave of pulse.  
independent variable and the dependent variable. As such, the multi-dimension regression model is defined as,  
\[y=B_{0}+B_{1}x_{1}+B_{2}x_{2}+\ldots+B_{k}x_{k}+e \tag{2}\]  
where \(e\) is the error. Each measured data is represented by \(i\). The regression model is  
\[y=B_{0}+B_{1}x_{i1}+B_{2}x_{i2}+\ldots+B_{k}x_{ik}+e_{i} \tag{3}\]  
\(B_{0}\), \(B_{1}\), \(B_{2}\)... \(B_{k}\) are obtained by using the matrix method,  
\[y=\left[\begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{n}\end{array}\right], \tag{4}\]  
\[X=\left[\begin{array}{cccc}1&x_{11}&x_{12}&\ldots&x_{1K}\\ 1&x_{21}&x_{22}&\ldots&x_{2K}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 1&x_{n1}&x_{n2}&\ldots&x_{nK}\end{array}\right], \tag{5}\]  
\[B=\left[\begin{array}{c}B_{1}\\ B_{2}\\ \vdots\\ B_{n}\end{array}\right], \tag{6}\]  
\[E\left[\begin{array}{c}e_{1}\\ e_{2}\\ \vdots\\ e_{n}\end{array}\right], \tag{7}\]  
\[y=X*B+E. \tag{8}\]  
Assuming the error value \(E\) is zero, then,  
\[\hat{y}=X*\hat{B}, \tag{9}\]  
\[X^{\prime}\Big{(}y-X\hat{B}\Big{)}=0, \tag{10}\]  
\[X^{\prime}y-X^{\prime}X\hat{B}=0, \tag{11}\]  
\[X^{\prime}y=X^{\prime}X\hat{B}. \tag{12}\]  
Finally,  
\[\hat{B}=\left(X^{\prime}X\right)^{-1}X^{\prime}y, \tag{13}\]  
Table 1 shows the parameters used in this study, and which have used to estimate the blood pressure, \(\hat{y}\) (systolic pressure and diastolic pressure). The root mean square error, \(E_{RMS}\), was used to evaluate the performance of the model,  
\[E_{RMS}=\left(\frac{1}{n}\sum\
olimits_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right) ^{2}\right)^{0.5}. \tag{14}\]  
where \(n\) is the number of data.