Introduction

Blood pressure (BP) is one of the significant vital signs of health. Oscillometric automated BP monitors are now easily and widely available, including in homes, offices and hospitals. The maximum amplitude algorithm (MAA) linearly estimates BP by approximating the mean arterial pressure (ABP) with the cuff pressure at which the maximum amplitude occurs and then using a fixed ratio of empirically derived coefficients [1]. The characteristic fixed ratio of these patients makes it possible to identify when the cuff pressure matches the SBP and DBP, respectively [2]. In fact, SBP is an arterial maximum blood pressure that occurs at the beginning of the cardiac cycle when the heart contracts, and DBP occurs when the heart relaxes between the heartbeat and the minimum blood pressure of the artery at the end of the cardiac cycle [3]. However, the MAA using fixed ratios makes acquiring reliable BP estimation difficult because individual subjects possess their own unique physiological characteristics [4]. To address the problem of fixed ratios in the estimation of BP, a neural network (NN) was performed [3]. However, this scheme did not enough the protocol of standard BP measurements [5]. To follow these recommendations, an advanced machine learning technique is still needed to estimate the BPs without characteristic ratios.

This paper proposes a fusion technique based on a support vector regression (SVR) [6], which can be adopted not only to classify some pattens, but also to estimate target values. Specifically, non-linear functions in SVR are mapped into the derived feature space of high dimensional kernel through linear learning machine, including using the features of the maximum margin algorithm [7]. Therefore, this method can represent the non-linear function between the feature vector collected from the oscillometric waveform (OMW) and the nurse’s BP. Moreover, a general approach is to use the SVR to produce an estimator using a training set with good results. However, there are some disadvantages to this approach. First, it is not simple to determine the best SVR estimator, and worthy information can be omitted when selecting one SVR estimator and discarding the others. Recently, deep networks have become a hot issue in machine learning and signal processing, but this technology requires a lot of time, cost and large data. Additionally, our input feature vectors, obtained from only five measurements per subject constitute a very small sample size, which is a critical weakness while using machine learning techniques, such as the SVR and FFNN. This shortcoming can lead to over-fitting [3]. In order to overcome these problems, we use the bootstrap-aggregation (bagging) technique to generate pseudo-features to estimate the SBP and DBP based on the SVR model [8]. Thus, the number of the pseudo-feature’s sample is drastically increased by employing the bagging technique. The bagging fusion estimator is a strong estimator through assembling a basic estimator that can perform better than random guessing [9], [10], which can effectively learn to mimic the non-linear model between the feature data acquired from the oscillometry and nurse’s BPs. The methodology proposed in the present study is one of the first to use the bagging ensemble regression model using the SVR technique for estimating the BPs. The proposed scheme is to increase accurate BP estimation and mitigate uncertainty BP estimation for a smart healthcare system [11], [12], [13].

This work has the succeeding augmentations and contributions when compared to [3]:

  • This paper introduces a new approach to obtain exact BP estimates using a relatively small sample data from the oscillometry using SVR-based fusion regression estimates.

  • This method can reduce overfitting and unstable estimation, such as large error on the mimic training feature set, through the SVR ensemble regression estimator.

Signal processing and feature extraction from the OMW signals are represented in Section “Signal processing and feature extraction from OMW signals”. The BP estimation based on the fusion technique based on SVR is introduced in Section “BP estimation based on the bootstrap aggregating technique based on SVR”. The results and discussion are presented in Section “Experimental results”. Finally, in Section “Conclusion”, this paper is concluded.

Signal processing and feature extraction from OMW signals

Data collection and preprocessing

This study was initially confirmed by the study committee and all volunteers agreed to provide information prior to the measurements based on the institutional research ethics committee’s BP research program. The BP measurements was collected from 85 volunteers without a history of cardiovascular disease between the ages of 12 and 80 years. Our BP measurement consists of five measurements for each volunteer, with a one minute break between measurements. The oscillometric BP measurement data of this research was provided by a company [16]. Oscillometric BP signals were obtained by detecting pressure pulsations in a subject’s cuff around the biceps or wrist as represented (in the second box on the Fig. 1). Details on OMW signal and BP measurement processing to estimate BP are provided in [16]. In this study, the OMW signals and envelopes were prepared for feature extraction, as shown in the step 3 on the left of Fig. 1. Afterward, the OMW signal is generated using the peak values of the oscillometric pulse. Next, we use the weighted median (WM) filter for smoothing the signal as shown in the step 4. We then use an asymmetric Gaussian fitting to acquire the smoothed envelope as shown in the step 5.

Features acquired from OMWs

In this subsection, we obtained the original features from the OMW signals and compared the target values utilizing the SVR technique. For more information about the feature vectors, please refer to the reference papers [4], [14] (Table 1).

Table 1 The list of abbreviation
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BP estimation based on the bootstrap aggregating technique based on SVR

As mentioned the first section, we use pseudo-features as explanatory data to predict the reference BPs as dependent variables as shown in the step 8. The block diagram of the proposed bagging regression technique based on SVR is shown Fig. 1. It comprises three parts such as left part (from step 1 to step 8), middle part (from step 9 to step 14), and right part (from step 15 to step 18).

figure a
Fig. 1
figure 1

Block diagram for BP prediction utilizing the SVR with the bagging fusion estimator. Left boxes (steps 1–8): Steps of BP measurement, preprocessing, feature extraction, and pseudo features generation. Middle boxes (steps 9–14): SVR training and SVM prediction. Right boxes (steps 15–18): Bagging ensemble to increase accurate and decrease uncertainty, where y(1) denotes the output vector of the first estimator, DT(1) denotes the mean vector of the first estimator, and K is the number of SVR estimator

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Support vector regression (SVR) estimator

Assume that one has a training data set {(x1,y1),...,(x l ,y l )}∈x i ×\(\mathbb {R}\), where x i denote the training data and y i denote the target BPs. The SVR is to find \( y = \langle \omega ,\phi (x)\rangle _{\mathcal {H}} + b\), where ω and ϕ(x) denote the vectors obtained from reproducing kernel Hilbert space \(\mathcal {H}\). Therefore, we can minimize the following risks:

$$ \mathcal{R}= \frac{1}{2}\parallel \omega \parallel^{2} + \mathcal{C}\sum\limits_{i = 1}^{l} \mathcal{L}(y_{i}, x_{i}, f), $$
(1)

where \(\mathcal {L}(y_{i}, x_{i}, f) \) denotes the 𝜖-loss function [6] given by

$$ \mathcal{L}(y_{i}, x_{i}, f) = \mid y - f(x)\mid_{\epsilon} = \max(0, \mid y - f(x)\mid - \epsilon) $$
(2)
$$ \underset{\omega,b,\xi,\tilde{xi}}\min\quad\frac{1}{2} \parallel \omega \parallel^{2} + \mathcal{C}\sum\limits_{i = 1}^{l} (\xi^{2} + \tilde{\xi}^{2}) $$
(3)
$$\text{subject to} \left\{ \begin{array}{lll} f(x_{i})-y_{i} & \leqslant\epsilon + \xi_{i},\quad & i = 1,...,l\\ y_{i}-f(x_{i})& \leqslant\epsilon + \tilde{\xi}_{i},\quad & i = 1,...,l\\ \xi_{i}, \tilde{\xi}_{i} \geqslant 0 & & \end{array}\right. $$

This problem can be solved by the Lagrangian theory as follows:

$$ \omega = \sum\limits_{i = 1}^{l}(\hat{\alpha}_{i} - \alpha_{i})\phi(x_{i}), $$
(4)

where \(\{\hat {\alpha }, \alpha \}\), i = 1,...,l denote the Lagrangian multipliers with respect to the constraints given in Eq. (4), and the solutions are given by

$$\begin{array}{@{}rcl@{}} &&\underset{\alpha_{i}, \hat{\alpha}_{i}}\max \sum\limits_{i = 1}^{l} y_{i}(\hat{\alpha}_{i} \,-\, \alpha_{i}) \,-\, \epsilon \sum\limits_{i = 1}^{l} (\hat{\alpha}_{i} \,+\, \alpha_{i}) \,-\, \frac{1}{2}\sum\limits_{i,j = 1}^{l}(\hat{\alpha}_{i} \,-\, \alpha_{i})(\hat{\alpha}_{j} \,-\, \alpha_{j})\\&&\left( K(x_{i},x_{j})+\frac{1}{\mathcal{C}}\zeta_{i,j}\right) \end{array} $$
(5)
$$\text{subject to} \left\{ \begin{array}{l} \sum\limits_{i = 1}^{l}(\hat{\alpha}_{i} - \alpha_{i}) = 0, \\ 0 \leqslant \alpha_{i}, 0 \leqslant \hat{\alpha}_{i},\quad i = 1,...,l, \end{array}\right. $$

where \(K(x_{i},x_{j}) = \langle \phi (x_{i}), \phi (x_{j}) \rangle _{\mathcal {H}}\) and ζi,j represents the Kronecker symbol. This problem can be resolved using some disassemble methods of the SVR [6]. To summarize, the SVR model can automatically achieve the relationship between the nurse’s BPs (RSBP and RDBP) and pseudo feature data, in the SVR training stage as shown in steps 9 to 11. The SBP and DBP in our SVR model are accurately acquired from the training stage as shown in steps 12 to 14.

Bagging technique

Beyond the bootstrap technique, ensemble-based algorithms using the bagging technique have been developed [8]. In this paper, we use the bagging approach to improve the performance of the SVR model. Specifically, we are interested in estimating the target BP’s function as given by \(f:\mathbb {R}^{\alpha } \rightarrow \mathbb {R}^{\beta }\) based on the data set {(X1,Y1),...,(X N ,Y N )}, where X are explanatory vectors and Y are response vectors. An estimated function \(\hat {f}(\cdot )\) can be built using the given data set. We therefore perform the same process even when the original data set would be changed, using the bagging technique. The fundamental concept of the bagging technique is to use the pseudo data generated from the original data set to obtain multiple estimates as \(\{\hat {f}_{1}(\cdot ),..., \hat {f}_{K}(\cdot )\}\) to address the unstable estimations such as the high variances of mean error or mean square error [8]. Making the ensemble estimators \(\hat {f}_{\varphi }(\cdot ) \) can be given as:

$$ \hat{f}_{\varphi}(\cdot) = \sum\limits_{k = 1 }^{K} \zeta_{k}\hat{f}_{k}(\cdot) $$
(6)

where \(\hat {f}_{k}(\cdot )\) denotes the estimated function acquired from the k th pseudo-data set; and ζ k is a weighted coefficient. In this paper, we can use the bagging technique to increase the reliability of the results and reduce the variance of the mean error in the SVR technique. In addition, the small number of sample problems acquired in each patient and the unstable performance of the SVR model can be solved simultaneously.

The target expectation is given by \(\mathbb {E}[{Y}|{X}]\), considering the proposed SVR model. Suppose that X = {x1,...,x N } and Y = {y1,...,y N } are random samples of the \({\mathbb {F}}\) distribution whose parameters are unknown {x,σ2}, respectively.

Therefore, we use the \(\hat {\mathbb {F}}\) employing sample parameters \((\bar {x}, \sigma ^{2}| {X})\), where the mean and variance are obtained from \(\mathbb {E}(x|{X}) \simeq \bar {x}= \frac {1}{n} {\sum }_{i = 1}^{n} x_{i}\) and \(\mathbb {E}(\sigma ^{2}|{X} ) \simeq \hat \sigma ^{2} = {\frac {1}{n-1} {\sum }_{i = 1}^{n}(x_{i} - \bar {x})^{2} }\), where \(\hat {\mathbb {F}}\simeq \mathcal {N}(\bar {x}, \sigma ^{2})\) is a normally distributed, referred to as the parametric bootstrap [15]. The parametric bootstrap is employed to generate the pseudo-feature vectors acquired from the mean and variance of the original for each subject. An identical process is utilized to generate the pseudo-features from Y = {y1,...,y N } to use the target BPs. Therefore, sufficient samples are prepared to predict the nurse’s BPs (SBP and DBP) as a training data set, and new features that do not participate in the training phase are utilized as test data.

Experimental results

BP monitors can pass standard protocols if the ME is less than 5 mmHg and the SDE is less than 8 mmHg [5]. The MEs of the proposed methodologies were readily computed by \((e_{i} = {t}_{i} - \hat {t}_{i})\). The ME and MAE were given as \((n^{-1} {\sum }_{i = 1 }^{n } e_{i}), (n^{-1} {\sum }_{i = 1 }^{n } \mid e_{i} \mid )\), respectively. The SDE values of ME and MAE were also easily computed using statistical equations.

Statistical analysis

The parameters of the proposed SVR ensemble algorithm are summarized in Table 2. Based on this layout, we performed a test to assess the performance of the SVR fusion technique when using equal conditions with a number of bootstrap replications (B(= 100)) for the pseudo-features. We confirmed that the SVR fusion technique achieved consistently good results. We showed that the SVR fusion estimator produced superior results compared to the FFNN, as demonstrated in Table 3. We evaluated the ME of SBP and DBP acquired employing the SVR fusion estimator with that of MAA[9] and FFNN [3].

Table 2 Parameter settings of the SVR fusion algorithm
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Discussion

The SDE values obtained by FFNN were achieved to be 7.61 and 6.79 mmHg, respectively, as presented in Table 3. The SDEs for the proposed SVR fusion model were improved by 2.18 and 1.58 mmHg compared to those of the MAA, as represented in Table 4. We confirmed differences of 0.26 and 0.32 mmHg in the SDE values, respectively, between the SVR fusion algorithm and the SVR single algorithm. Additionally, we compared to conventional FFNN algorithm. According to the BHS protocol, the SVR fusion methodology acquired grades of A and A for the SBP and DBP, respectively. The results of the SVR fusion estimator were 66.82 % (≤ 5 mmHg), 88.00 % (≤ 10 mmHg), and 95.53 % (≤ 15 mmHg) for the SBP in the given experimental test, and 74.82 % (≤ 5 mmHg), 94.12 % (≤ 10 mmHg), and 98.82 % (≤ 15 mmHg) for the DBP in the experimental test, as shown in Table 4. Conclusively, we confirmed that the SVR fusion estimator outperformed the conventional models, as indicated in Tables 3 and 4. This implies that enough pseudo-features are a critical factor for enhancing the generalization ability of the SVR fusion estimator. The results indicated that the estimated blood pressure values obtained by the SVR ensemble estimator were close to the reference blood pressures (SBP and DBP). The probabilities of the BHS standard protocol [17,18,19,20,21] represented that the proposed SVR fusion algorithm offered accurate BP estimates compared with the FFNN estimator as shown Figs. 2 and 3.

Table 3 ME and SDE relative to the nurse’s measurements acquired from the conventional MAA, FFNN, and SVR [6], and using the proposed SVR ensemble (φ) estimator
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Table 4 The conventional MAA, FFNN, SVR [6] and proposed SVR fusion estimators show the performance results of (5 × 85 = 425) algorithms based on the BHS protocol
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Fig. 2
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The Bland-Altman plots [5]

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Fig. 3
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The Bland-Altman plots [5]

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Conclusion

In conclusion, the proposed technique incorporates an SVR fusion estimator and offers a method that can mitigate uncertainty for oscillometric BP measurements using the bootstrap ensemble technique. The results demonstrate that this methodology improves the accuracy of BP estimation. The proposed strategy can be used for various kinds of oscillation phenomena such as a speech, a photoplethysmography, and an electrocardiography signals.