Unsupervised ensembling of multiple software sensors: a new approach for electrocardiogram-derived respiration using one or two channels

06/23/2020 ∙ by John Malik, et al. ∙ Duke University 0

While several electrocardiogram-derived respiratory (EDR) algorithms have been proposed to extract breathing activity from a single-channel ECG signal, conclusively identifying a superior technique is challenging. We propose viewing each EDR algorithm as a software sensor that records the breathing activity from the ECG signal, and ensembling those software sensors to achieve a higher quality EDR signal. We refer to the output of the proposed ensembling algorithm as the ensembled EDR. We test the algorithm on a large scale database of 116 whole-night polysomnograms and compare the ensembled EDR signal with four respiratory signals recorded from four different hardware sensors. The proposed algorithm consistently improves upon other algorithms, and we envision its clinical value and its application in future healthcare.

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1. Introduction

Respiration is the primary physiological activity of the human body, which is the process of converting glucose to energy on the cellular level. Breathing, on the other hand, is the physical process that transfer gases between cells and the environment. In the clinic, the respiratory signal (or breathing signal) is a broad term that refers to any signal containing information related to the breathing process. Respiratory signals provide clinicians with important information concerning the breathing process, including tidal volume [15], respiratory rate, and the variability of the breathing pattern [3]. The respiratory rate quantity can be accessed by counting the number of breaths a patient takes in a one-minute period [31, 34]. On the other hand, effectively assessing breathing pattern variability requires more signal processing [4]. These quantities can be used to monitor respiratory health or to predict respiratory failure [12].

On the respiratory signal recording side, the gold standard amongst respiratory signals is the spirogram collected by the spirometry. However, spirometry is uncomfortable and not ideal for long-term monitoring [13]. Besides the spirometer, a variety of hardware and software sensors for measuring respiration have been proposed [23]

. Compared to the spirogram, the information obtained from these sensors might be limited, and obtaining a reliable respiratory signal is challenging. In this paper, we focus on the respiratory signal that can be extracted from the electrocardiographic (ECG) signal, which is an ensembled software methodology for estimating breathing activity through easy-to-use, mobile-capable, and non-invasive ECG sensors. It is based on the fundamental physiological fact – when the patient inhales (resp. exhales), transthoracic electrical impedance increases (resp. decreases), and the amplitude of the ECG signal decreases (resp. increases) and the ECG morphology changes.

Deriving the respiratory signal from a multiple- or single-channel ECG has been attempted using a number of methods [25, 17, 6]

. Traditionally, the EDR signal is obtained by interpolating the time series of RS amplitudes as follows:

(1)

where is the discrete-time ECG signal having length and a sampling rate of Hz, is the sample index of the -th R peak, and is the sample index of the subsequent S peak. Another method involves estimating the area of each QRS complex [6, Chapter 8]

. Principal component analysis (PCA), independent component analysis, and kernel principal component analysis (kPCA) have also been used to obtain an EDR signal

[19, 16, 39, 37]. Recently, an EDR algorithm based on the diffusion maps [7] algorithm was proposed [22].

While many EDR approaches exist, the superiority of one particular technique is debated. For example, one comparison article advocates for the method based on RS amplitudes [38], while another advocates for the method based on measuring QRS areas [29]. Since it is difficult to conclude which EDR algorithm is optimal, it is natural to consider fusing all such sensors together; that is, combining available EDR algorithms to obtain a breathing activity signal that is generally better than existing EDR signals. In other words, we can view each EDR algorithm as a software sensor

for the breathing activity, and try to combine these software sensors to obtain a better one. This idea, which is similar to the concept of ensemble learning in machine learning

[30], has lead to this work.

In this work, we provide two main contributions. First, we introduce and test multiple nonlinear sensor fusion approaches to deriving the respiratory signal from a two-channel ECG signal. Second, we show that the existing approaches to estimating the respiratory signal from a single- or two-channel ECG signal can be combined, or ensembled, to enhance the quality of the breathing activity estimation. In other words, a multitude of initial EDR estimates, viewed as the outputs of different software sensors measuring breathing activity from the ECG signal, are ensembled to yield a respiratory signal that is of superior quality. We refer to the proposed EDR signal as the ensembled EDR.

2. Methods

We provide a detailed description of our proposed ensembled EDR algorithm that extracts breathing activity from a two-channel ECG signal. The flowchart of the algorithm is shown in Figure 1. Implicit in the description of this algorithm is the corresponding algorithm that should be employed when only a single-channel ECG signal is available. We will make the distinction between the single- and two- channel algorithms clear when necessary.

FirstECG Lead

Single-channelEDR Estimates

Dual-channelEDR Estimates

Single-channelEDR Estimates

SecondECG Lead

Ensembling

Figure 1. We visually portray the main steps in the proposed ensembed EDR algorithm, in which multiple EDR signals (estimated using diverse methods) are fused to form a stronger one. In the middle, we show the individual EDR estimates which, while different, share the information that can be extracted by the ensembling method. Our algorithm can achieve a superior estimate when one or two ECG leads are available. When there are two ECG leads, various estimates for the respiratory signal are made on each lead individually and on both leads simultaneously. All of the estimates are then ensembled to obtain the final EDR signal. We note that this method can in theory be extended when more ECG leads are available.

2.1. Step 1: Pre-processing

The raw ECG signal is first upsampled to a sampling rate of Hz. Measurement noise and powerline interference in the ECG signal are surpressed by applying a bi-directional, third-order Butterworth lowpass filter with a cutoff frequency of Hz. Baseline wandering is estimated and subsequently removed by applying a median filter with a window size of ms to each channel of the ECG signal. This step also moderately suppresses the P and T waves. We detect the QRS complexes in each channel of the ECG signal by applying a standard high-accuracy QRS detector [24]. We apply a standard beat matching algorithm [10]

to ensure that all of the QRS complexes detected in the first channel are also present in the second channel, and vice versa; the extraneous detections are removed. When using only one ECG channel, we perform beat-matching after applying two different QRS complex detection algorithms to the same ECG channel. Denote the pre-processed two-channel ECG signal as a pair of vectors

, where is the number of samples, Hz is the sampling rate of the signal, and is the duration of the recording in seconds. Suppose there are detected R peaks in the ECG signal. Let denote the location in samples of the th R peak in the first channel, and let denote the location in samples of the th R peak in the second channel. For the purpose of analyzing the traditional EDR signal, the S peak following each R peak is determined by

(2)

where the window length of samples, corresponding to a  ms duration, is chosen according to normal electrophysiology. By defining a window around each R peak that extends samples (ms) to the left and samples (ms) to the right, we are able to capture the entire QRS complex, recalling that the P and T waves have been suppressed. For each channel ( and ), we collect the segments containing QRS complexes and stack them as rows in a matrix:

(3)

where the resulting matrix is , where . Note that the row ordering of encodes the temporal information.

2.2. Step 2: Estimating the respiratory signal

We build a variety of estimates for the respiratory signal; these estimates, resulting from several techniques, are viewed as software sensors for the breathing activity. When only one ECG channel is available, we consider the traditional approach based on RS amplitudes (1), the geometric approach (PCA), and the manifold learning approach (diffusion maps) to obtain a pool of EDR signals. When two ECG channels are available, we apply various sensor fusion approaches, including canonical correlation analysis [18], an extension [33] of alternating diffusion [20, 36], and another variant of alternating diffusion based on the dynamic Laplacian [14]. Subsequently, these estimated respiratory signals from various software sensors are fused to obtain a new EDR signal, the ensembled EDR.

2.2.1. Traditional approach

The first estimate for the respiratory signal is the traditional EDR signal. We set

(4)
(5)

so that after piecewise cubic spline interpolation at a sampling rate of Hz, the assigned points and yield interpolated vectors . This interpolation is applied to all the subsequent EDR estimates, and thus, for brevity, is omitted.

2.2.2. Geometric approach

The next two estimates are obtained via principal component analysis (PCA). Let be the top five principal components of the matrix

(that is, the top five eigenvectors of the corresponding

covariance matrix), and let be the top five principal components of the matrix . We then set

(6)
(7)

Details about this method can be found in [19, 28]. We mention that the number five is chosen in an ad hoc way. A different number of principal components can be considered.

2.2.3. Manifold learning approach

Next, we apply a nonlinear manifold learning method called the diffusion maps algorithm [7], which can be viewed as a generalization of the kernel PCA approach [39]. We build self-tuning affinity matrices by setting W_l(i, j) = 12 exp[ - ∑_k=1^p (Xl((i, k) - Xl((j, k) )2σi(l) ] + 12exp[ - ∑_k=1^p (Xl((i, k) - Xl((j, k) )2σj(l) ], where is the median of the squared Euclidean distances from the -th row of to all other rows. We then perform -normalization with to mitigate any non-uniform densities (see [7] for details). We calculate

(8)

where is a diagonal matrix satisfying . We then calculate the isotropic diffusion kernel by setting

(9)

where is a diagonal matrix satisfying . We obtain the top five non-trivial eigenvectors of , and we set to be the vector

(10)

Finally, we set

(11)
(12)

Again, we mention that a different number of eigenvectors can be considered.

2.2.4. Linear sensor fusion approach

First, let denote the top five left-singular vectors of the matrix , and let denote the top five right-singular vectors of the same matrix [18]. We set

(13)

for all and .

2.2.5. Nonlinear sensor fusion approaches

To perform the extended alternating diffusion algorithm [33], we calculate

(14)
(15)

Let be the eigenvectors of the anti-symmetric matrix

that correspond to the five non-conjugate eigenvalues of largest magnitude. Set

(16)

for all and , where and denote the real and imaginary parts, respectively. Let be the top five eigenvectors of the symmetric matrix , and set

(17)

for all and .

Finally, we introduce a decomposition of the discrete dynamic Laplacian [14]. We let be the top five non-trivial eigenvectors of the operator

(18)

and set

(19)

for all and .

2.3. Step 3: Ensembling the estimated respiratory signals

Prior to combining all estimates for the respiratory signal, we normalize them. For each and each estimate , set

(20)

This procedure corresponds to locally -scoring the signal.

We proceed with ensembling the estimates using a method that can be extended to include any type of EDR signal. Consider the initial matrix , whose columns are the EDR estimates. Then, we compute the time-lagged representations , given by

(21)

for all , , and . We define a matrix , whose rows are given by

(22)

for all . Note that has columns. Let be the top left-singular vector of . We view this time series as the final estimate for the respiratory signal, the ensembled EDR. Note that the first entry represents the estimated respiratory volume at time seconds; to compensate for this lag, we add null values at the beginning of the signal , whose effective sampling rate is Hz.

The rationale underlying the use of singular value decomposition (SVD) for fusing the estimates is as follows. Geometrically,

represents the direction (in the high-dimensional space ) at which the power of all of the estimates is maximized. In other words, represents the common information between all of the estimates.

In some cases, only one ECG lead is available. In these cases, we do not have access to any of the estimates of the respiratory signal that require two ECG channels. However, we can still use the procedure described above, only now , and subsequently , consist only of the signal-channel estimates. The rest of the steps remain unchanged.

2.4. Material

The ECG signals featured in our experiment are from a set of 116 standard overnight polysomnograms which were recorded to confirm the presence of sleep apnea syndrome in subjects suspected of sleep apnea at the sleep center in Chang Gung Memorial Hospital (CGMH), Linkou, Taoyuan, Taiwan. The Institutional Review Board of CGMH approved the study protocol (No. 101-4968A3). All recordings were acquired on the Alice 5 data acquisition system (Philips Respironics, Murrysville, PA). All subjects had an apnea-hypopnea index less than 5. We use both leads of the ECG recording, which, before upsampling, were sampled at 200 Hz. To evaluate the effectiveness of our estimate for the respiratory signal, we compare it to the FLOW, CFLOW, THO, and ABD signals, where FLOW is the patient flow signal recorded from the thermistor, CFLOW is the patient flow signal recorded from the nasal cannula, THO is recorded from the chest band with the piezosensor, and ABD is recorded from the abdominal band with the piezosensor.

2.5. Evaluation

To evaluate the performance of our algorithm and compare it to the performance of other algorithms, we employ the following procedure. We randomly select eight non-overlapping 30-minute segments from each subject’s polysomnogram. The total number of 30-minute segments is 928. Using the two (or one, as appropriate) ECG leads, we build our proposed ensembled EDR on each 30-minute segment. Let denote the ensembled EDR for one 30-minute segment, and let denote either the FLOW signal, the CFLOW signal, the THO signal, or the ABD signal. For each integer lag parameter in the range , we consider the shifted time series

(23)

and calculate the absolute value of the Pearson correlation coefficient between the first two minutes of and (ignoring any null values in ). Our final measure of similarity between and is

(24)

The parameter allows us to account for any delay between the ECG-derived respiratory signal and the respiratory signal as measured through the various sensors. The metric is inherited from a previous work which examined the use of kernel PCA for building an EDR signal [39].

Inspired by a previous work [2]

, we introduce a second evaluation metric to quantify the extent to which our ensembled EDR signal can accurately measure

time-varying respiratory rate. Effectively, it is a comparison between the time-frequency representations (TFR) of and , where is maximum argument of (24). We use the de-shape synchrosqueezing transform (dsSST) [21] to calculate both TFRs. The dsSST is designed to remove harmonics in the spectrogram that result from the non-sinusoidal nature of the breathing pattern. (These harmonics are not relevant when assessing instantaneous respiratory rate.) We use a window size of 200 samples, a Gaussian window with bandwidth , a soft log parameter of 0.03, a hop of , and DFT points in the dsSST. The columns of both TFRs are -normalized and compared using the 1-Wasserstein distance [8, Equation 4.3], yielding the metric . Note that the 1-Wasserstein distance is the distance between cumulative density functions.

To examine how the signal length affects the performance of our new ECG-derived respiratory signal, we repeat the above experiment using the first , , and minutes of each selected -minute segment. We remark that is calculated using only the first two minutes of and because this way, we can accurately assess the effect of using different signal lengths to estimate the respiratory signal.

3. Results

We begin by examining the case when only one ECG lead is available. In this case, we compare the signal obtained by fusing 11 single-channel respiratory signal estimates to the traditional EDR signal and to the EDR signal obtained via PCA (the first principal component). In Table 1, these signals are compared to the accompanying FLOW, CFLOW, THO, and ABD signals in the polysomnogram. Higher values indicate a more accurate recovery of the respiratory signal. We carry out four assessments on ECG signals of lengths , , , and minutes. (Note that the -minute signal is the first two minutes of the corresponding -minute signal, and so on.) The ensembled EDR signal in the one-channel scenario (Table 1) obtains a higher index compared with the other individual methods. The evaluation by the OT distance, the index, is shown in Table 2. The smaller OT distance indicates a better recover of the respiratory signal in the sense of instantaneous respiratory rate recovery. It is shown that the proposed ensembled EDR consistently performs better.

Next, we examine the case when two ECG leads are available. In this case, the signal is obtained by fusing 52 estimates for the respiratory signal made on either one of the two leads or on both leads simultaneously (see Section 2.3). Of these 52 preliminary EDR signals, a number of them are obtained using novel methods from the field of nonlinear sensor fusion. In Table 3, we compare both and these novel two-channel estimates with the accompanying FLOW, CFLOW, THO, and ABD signals. The ensembled EDR signal in the two-channel scenario (Table 3) obtains a significantly higher value compared with all other one- and two-channel methods. The evaluation by the index is shown in Table 4, and the proposed ensembled EDR consistently performs better.

We carry out the following statistical tests to examine how the length of the signal affects the optimality of our ensembled estimate for the respiratory signal. We focus on the two-channel ensembled EDR algorithm only. The results of this test are shown in Table 5. For each target respiratory signal (FLOW, CFLOW, THO, or ABD) and each evaluation metric ( or

), we carry out Friedman’s test to evaluate the null hypothesis that the signal length has no effect on the evaluation metric (for the chosen target respiratory signal). The alternative hypothesis is that the signal length has a statistically significant effect on the evaluation metric (for the chosen target respiratory signal). If the null hypothesis is rejected, we carry out an additional

multiple comparisons test (using Tukey’s honestly significant difference criteria) to identify which signal lengths were optimal. After the procedure, we found that using signals minutes in length yielded better values when the target respiratory signal was either the FLOW, THO, or ABD signal. Moreover using signals of minutes in length yielded better values when the target respiratory signal was CFLOW. In all other cases, no statistically significant effects were observed.

FLOW
Traditional
EDR ()
PCA ()
Diffusion
Maps ()
Single-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

CFLOW
Traditional
EDR ()
PCA ()
Diffusion
Maps ()
Single-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

THO
Traditional
EDR ()
PCA ()
Diffusion
Maps ()
Single-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

ABD
Traditional
EDR ()
PCA ()
Diffusion
Maps ()
Single-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

Table 1. Performance of the proposed single-channel ensembled ECG-derived respiratory (EDR) signal evaluated by the index. Entries in the table are the mean standard deviation of all indices. A statistically significant improvement by the single-channel ensembled EDR (as evaluated by a one-sided Wilcoxon signed rank test with significance level ) is indicated by the marker .
FLOW
Traditional
EDR ()
PCA ()
Diffusion
Maps ()
Single-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

CFLOW
Traditional
EDR ()
PCA ()
Diffusion
Maps ()
Single-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

THO
Traditional
EDR ()
PCA ()
Diffusion
Maps ()
Single-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

ABD
Traditional
EDR ()
PCA ()
Diffusion
Maps ()
Single-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

Table 2. Performance of the proposed single-channel ensembled ECG-derived respiratory (EDR) signal evaluated by the OT distance, the index. Entries in the table are the median median absolute deviation of all values. A statistically significant improvement by the single-channel ensembled EDR (as evaluated by a one-sided Wilcoxon signed rank test with significance level ) is indicated by the marker .
FLOW
Canonical
Correlation ()
Alternating
Diffusion ()
Dynamic
Laplacian ()
Two-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

CFLOW
Canonical
Correlation ()
Alternating
Diffusion ()
Dynamic
Laplacian ()
Two-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

THO
Canonical
Correlation ()
Alternating
Diffusion ()
Dynamic
Laplacian ()
Two-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

ABD
Canonical
Correlation ()
Alternating
Diffusion ()
Dynamic
Laplacian ()
Two-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

Table 3. Performance of the proposed two-channel ensembled ECG-derived respiratory (EDR) signals evaluated by the index. Entries in the table are the mean standard deviation of all indices. A statistically significant improvement by the single-channel ensembled EDR was observed in all cases (as evaluated by a one-sided Wilcoxon signed rank test with significance level ).
FLOW
Canonical
Correlation ()
Alternating
Diffusion ()
Dynamic
Laplacian ()
Two-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

CFLOW
Canonical
Correlation ()
Alternating
Diffusion ()
Dynamic
Laplacian ()
Two-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

THO
Canonical
Correlation ()
Alternating
Diffusion ()
Dynamic
Laplacian ()
Two-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

ABD
Canonical
Correlation ()
Alternating
Diffusion ()
Dynamic
Laplacian ()
Two-channel
ensemble
2 minutes
5 minutes
15 minutes
30 minutes

Table 4. Performance of the proposed two-channel ensembled ECG-derived respiratory (EDR) signals evaluated by the OT distance, the index. Entries in the table are the median median absolute deviation of all indices. A statistically significant improvement by the two-channel ensembled EDR (as evaluated by a one-sided Wilcoxon signed rank test with significance level ) is indicated by the marker .
FLOW Null
CFLOW Null
THO Null
ABD Null

Table 5. Examining the effects of signal length on the performance of the two-channel ensembled EDR signal. When Friedman’s test indicates the presence of signal length effects, these effects are investigated using Tukey’s honestly significant difference criteria. The notation means that using signals of length minutes was significantly better than using signals of length minutes. “Null” means that Friedman’s test did not allow a rejection of the null hypothesis.

In Figure 2, we show a short segment of the ensemble of EDR signals next to the accompanying FLOW, CFLOW, THO, and ABD signals. In Figure 3, we show a comparison of the traditional EDR signal obtained from the first ECG lead, the EDR signal obtained from the first ECG lead using the DM algorithm, the single-channel ensembled EDR signal obtained from the first ECG lead, and the two-channel ensembled EDR signal. Compared with the EDR signal from the DM algorithm and the ensembled EDR signal, It is clear that the traditional EDR signal oscillates in an irregular way; specifically, the oscillatory pattern changes from one to another.

Figure 2. An unsupervised ensembling of EDR signals produces one which accurately measures the respiratory cycle. From top to bottom, we show: the ensembled EDR signal; the FLOW signal; the CFLOW signal; the THO signal; and the ABD signal.
Figure 3. An unsupervised ensembling of EDR signals, the proposed ensembled EDR algorithm, produces one which accurately measures the respiratory cycle. From top to bottom, we show: the first ECG lead; the second ECG lead; the traditional EDR signal obtained from the first ECG lead; the EDR signal obtained from the first ECG lead using the DM algorithm; the single-channel ensembled EDR signal obtained from the first ECG lead; the two-channel ensembled EDR signal; and the CFLOW signal.

4. Discussion

In this work, we demonstrate that ensembling multiple EDR signals can lead to a better estimate for the respiratory signal, which we call the ensembled EDR signal. We mention that the ensembling idea for the EDR signal could be extended to other biomedical time series. We also demonstrate the novel application of several sensor fusion techniques for extracting the amplitude modulation of a two-channel ECG signal, including canonical correlation analysis, alternating diffusion maps, and the discrete dynamic Laplacian. In the single-channel case, the traditional EDR signal is fused with the EDR signal obtained via PCA and the EDR signal obtained via the diffusion maps algorithm to achieve an estimate for the respiratory signal which exceeds all individual estimates. In the two-channel case, the traditional EDR signals from both leads are fused with 50 other EDR signals to obtain one final, superior estimate (see Figure 1).

In this work, we use manifold learning methods such as the diffusion maps algorithm to recover the topology of the manifold which underlies the set of observed QRS complexes. When the manifold is nonlinear, one significant challenge to extracting respiratory dynamics is that this information can be spread non-linearly across multiple eigenvectors of the graph Laplacian. To be specific, it is not guaranteed that the first non-trivial eigenvector will contain (or be the sole container of) the respiratory information. See [22] for an extensive discussion. This problem is also present when we attempt to use linear methods such as PCA to recover a nonlinear manifold. However, when the underlying manifold of QRS complexes is low-dimensional (an assumption we make based on a priori physiological knowledge), an embedding of that manifold can be constructed using a small number of eigenvectors of the graph Laplacian [27], where the spectral convergence in the sense with a convergence rate is provided in [9]. We accept that there will be a distortion of the underlying manifold incurred when we use only five eigenvectors of the graph Laplacian to recover this manifold, but we can guarantee that somewhere in these chosen eigenvectors lies the respiratory information. We furthermore note that this distortion is preferable to allowing additive noise near the manifold to contaminate our embedding [11]. One issue that we do not address in this work is the linearity of the interpolation step. In principle, the interpolation should be done along the surface of the manifold. Algorithms for accomplishing this step will be explored in future work.

We see that as the length of the ECG signal increases past five minutes, the values for most methods decrease and the values for most methods decrease. The tests shown in Table 5

support this observation. While asymptotical analysis of the diffusion and other such operators guarantees an improved recovery of the manifold underlying the set of QRS complexes when the number of observed QRS complexes increases, we believe that the decrease when using signals longer than five minutes is due to non-stationarity in the time series. The extent to which stationarity (or a lack of stationarity) impacts the viability of the wave-shape oscillatory model for modeling dynamics in the ECG signal has not yet been explored

[22].

In our algorithm, all parameters were chosen ad hoc and without any explicit optimization or search. In particular, there is freedom to experiment with different bandwidths for the Gaussian kernel used in our algorithm, or to use the previously proposed selection criteria [39]. An optimization approach may yield stronger results but also may lead to over-fitting at the database level. As is illustrated in Figure 1, different EDR signals might be out of phase. Such out-of-phase behavior might downgrade the SVD performance. It is possible to consider the synchronization idea like graph connection Laplacian [35]

to synchronize the global phase before applying SVD. However, due to the time-varying nature of the respiratory rate, and probably other physiological interactions among ECG and respiratory dynamics, a direct application of this global phase synchronization idea does not work well. A systematic study of this issue is of an independent interest, and will be explored in the future work.

For validating our ensembled EDR signal, we propose the use of four different signals (FLOW, CFLOW, THO, ABD) because of the indirect way all of these signals are able to monitor breathing activity and we do not have access to ground truth breathing information. While physiologically these channels all capture the same breathing activity, by examining the values between each of the four respiratory signals (see Table 6), we can quantitatively confirm that they contain common information. Moreover, the values between the different respiratory signals give a standard for what we could hope to achieve using the ECG signal. The evaluation metric has been inherited from a previous work which introduced the use of kernel PCA for obtaining an EDR signal [39]. One other interesting observation in Tables 1 and 3 is that in the single-channel case, the respiratory signal that is most highly correlated with our new EDR signal is the ABD signal.

CFLOW THO ABD
FLOW
CFLOW
THO

Table 6. Four respiratory signals were used for validation. We show the values obtained between each pair of signals across all of the segments included in our study. We consider signals of length 2 minutes.

Further studies to explore the practical value of sensor fusion EDR is warranted, in particular the application in remote monitoring or mobile healthcare via smartphone app. Nowadays, as the advance of sensor technologies, there are more and more easy-to-use, mobile capable, or less-invasive sensors. While their qualities are tended to be questioned, the ensemble idea considered in this paper might be helpful to improve the overall quality of the collected signal. For example, it is well known that the photoplethysmogram not only reflects hemodynamic status but also the breathing activity [1]. Thus, we may consider combining respiratory information extracted from the ECG or photoplethysmogram [5], or even surveillance video, to further improve the quality of the obtained respiratory signal. On the other hand, the proposed ensemble EDR may contain richer respiratory information that could be better extracted by the advanced signal processing tools; for example, we may apply the time-frequency analysis [40] to more accurately extract the momentary respiratory rate, or other respiratory quantities.

We mention that there exists other approach that units a pool of estimates for the breathing activity made on a one- or two-channel ECG signal. Specifically, time-frequency analysis has been proposed for the task of sensor fusion [2]; the spectrograms of different EDR estimates are averaged to obtain a robust estimate for the respiratory rate. This method has merit for respiratory rate estimation, but its performance of recovering respiratory signal is not clear due to its lossy nature of spectrogram fusion.

This work has several limitations. Since the kernel methods we employ (such as the diffusion maps algorithm) involve computing the top eigenvectors of matrices (where is the number of heartbeats), the computational time required by our algorithm is high and increases as the length of the signal increases. For very long ECG signals where additionally non-stationarity will come into play, we suggest estimating the respiratory signal over short windows of minutes. The recently proposed algorithm, Roseland (Robust and Scalable Embedding via LANmark Diffusion) [32], could be considered to speed up those diffusion-based algorithms. A second limitation of our study is that we do not explicitly handle premature ventricular contractions, a significant obstacle when estimating the amplitude modulation of the ECG signal. These morphologically distinct beats should be removed by an algorithm such as [26] before running any EDR algorithm. We do not detect and remove motion artifacts from the ECG signals considered in this study, but our novel normalization procedure is designed to minimize their effect. Furthermore, we do not assess the signal quality of the accompanying FLOW, CFLOW, THO, and ABD signals before using them to assess the various EDR signals considered in this work. As far as we know, suitable signal quality indices for such signals have not yet been developed. Finally, we only consider sleeping subjects without sleep apnea in this work, and we cannot confirm the effectiveness of our algorithm on awake subjects or on sleeping subjects with sleep apnea. Our future work will include validating the proposed algorithm in a more general population.

In summary, we proposed and verified an ensembling approach to unite the information in different EDR algorithms. The resulting ensembled EDR signal was demonstrated to be of superior quality. We envision its clinical value and its application in clinical medicine and healthcare.

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