# Effects of Missing Data on Heart Rate Variability Metrics

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Aims of the Study

## 2. Materials and Methods

#### 2.1. Simulation of Missing Beats

#### 2.2. Apple Watch Dataset

#### 2.3. Missing Data Detection

#### 2.4. Correction Methods

#### 2.5. HRV Metrics

- Time domain: Mean heart rate (MHR), standard deviation of the normal-to-normal interval (SDNN) and root mean square of successive differences (RMSSD), as described in [1].
- Frequency domain: LF and HF powers (${P}_{LF}$,${P}_{HF}$); LF power measured in normalized units (${P}_{LFn}$); and ${P}_{LF}$/${P}_{HF}$ ratio. Only relative errors of ${P}_{LF}$ and ${P}_{HF}$ are presented, as the other two are derived from them. While all subjects are included when measuring relative errors, not all of them could be included when measuring the ability to distinguish sympathovagal balance. For this comparison, only subjects with respiratory rates above the classic LF band (>0.15 Hz) were selected, thus allowing a correct frequency component separation [34]. Therefore, simulation dataset is reduced from 16 to 9 subjects (age $28.3\pm 2.6$ years, 5 males). This selection only applies when comparing metrics in the frequency domain. No selection is made in the Apple Watch dataset. In addition, respiratory rate does not exceed 0.4 Hz—the classic HF band upper limit—in any case.Spectral estimation is performed via Fast Fourier Transform (FFT) and Lomb’s methods. FFT estimations are made on the evenly-sampled instantaneous heart rate signal, $r\left(t\right)$, obtained from the IPFM model [17]. This model assumes the ANS modulates the sinoatrial node by a band-limited zero-mean signal [35]. In [36], it is shown that spectra derived from $r\left(t\right)$ are a more accurate estimator for HRV than spectra derived from evenly-sampled interval series, avoiding spurious components and low-pass filtering effects. Welch’s method is used for periodogram averaging using 60 s Hamming windows with 50% overlap. For 120 s signals, three periodograms are averaged. Powers are computed using trapezoidal integration and classic windows (0.04–0.15 Hz for LF and 0.15–0.4 Hz for HF). This FFT-based approach has been tested using both model-based and gap-filling correction.On the other hand, Lomb’s periodograms can be computed from unevenly spaced signals, even in the presence of missing beats. Therefore, this method has been tested both using $\mathcal{OR}$ and gap-filling correction. It is demonstrated that the estimates on the heart rate representations are more accurate than on the beat interval representations [36]; therefore, Lomb’s periodograms are computed on the inverse interval function$${d}_{IIF}\left({t}_{k}\right)=\sum _{k=1}^{K}\frac{1}{({t}_{k}-{t}_{k-1})}\delta (t-{t}_{k})$$
- Poincaré plots: SD1, SD2, SD1/SD2, ellipse area ($S=\pi \xb7SD1\xb7SD2$), mean distance to the ellipse centroid (Md) and standard deviation to the ellipse centroid (Sd) have been computed using the ellipse fitting method [37]. As S and SD1/SD2 are computed from SD1 and SD2, relative errors are not shown for these metrics. The reliability of Poincaré plots in ultra-short term segments—less than 5 min, as this case—has been demonstrated recently [38].

#### 2.6. Statistical Analysis

## 3. Results

#### 3.1. Time-Domain Metrics

#### 3.2. Frequency-Domain Metrics Computed via FFT

#### 3.3. Frequency-Domain Metrics Computed via Lomb’s Method

#### 3.4. Poincaré Plots

## 4. Discussion

#### Limitations

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HRV | Heart rate variability |

PRV | Pulse rate variability |

ANS | Autonomic nervous system |

ECG | Electrocardiography |

PPG | Pulse photoplethysmography |

IBI | Inter-beat interval |

SNR | Signal-to-noise ratio |

OD | Outlier detection |

OR | Outlier removal |

L | Linear (correction method) |

NL | Non-linear (correction method) |

M | Model-based (correction method) |

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**Figure 1.**Example of simulation with a segment of 40 beats. Deleted beats are displayed in red. (

**a**) Random distributed missing beats, $p=0.25$. (

**b**) Bursts of missing beats. The elements at the ends (green) cannot be deleted.

**Figure 2.**Process flow. $\mathcal{OD}$ = Outlier Detection; $\mathcal{OR}$ = Outlier Rejection; $\mathcal{L}$ = Linear; $\mathcal{NL}$ = Non-Linear; $\mathcal{M}$ = Model-based.

**Figure 5.**

**Relax (green)/stress (blue) discrimination of time-domain metrics from Apple Watch dataset.**(

**a**) MHR. (

**b**) SDNN. (

**c**) RMSSD. *: Significant differences ($p<0.05$) between relax and stress groups. **: Significant differences ($p<0.001$) between relax and stress groups.

**Figure 6.**

**Coverage of frequency-domain metrics computed via FFT from Apple Watch dataset.**(

**a**) ${P}_{LF}$. (

**b**) ${P}_{HF}$.

**Figure 7.**

**Relax (green)/Stress (blue) discrimination of frequency-domain metrics computed via FFT from Apple Watch dataset.**(

**a**) ${P}_{LF}$. (

**b**) ${P}_{HF}$. *: Significant differences ($p<0.05$) between relax and stress groups. **: Significant differences ($p<0.001$) between relax and stress groups.

**Figure 8.**

**Coverage of frequency-domain metrics computed via Lomb’s method from Apple Watch dataset.**(

**a**) ${P}_{LF}$. (

**b**) ${P}_{HF}$.

**Figure 9.**

**Relax (green)/Stress (blue) discrimination of frequency-domain metrics computed via Lomb’s method from Apple Watch dataset.**(

**a**) ${P}_{LF}$. (

**b**) ${P}_{HF}$. **: Significant differences ($p<0.001$) between relax and stress groups.

**Figure 11.**

**Relax (green)/stress (blue) discrimination of Poincaré metrics from Apple Watch dataset.**(

**a**) SD1. (

**b**) SD2. (

**c**) S. (

**d**) Md. (

**e**) Sd. *: Significant differences ($p<0.05$) between relax and stress groups. **: Significant differences ($p<0.001$) between relax and stress groups.

**Table 1.**

**Relative error (%) of time-domain metrics. (a)**Scattered missing beats.

**(b)**Bursts. †: Significant differences ($p<0.05$) between $\mathcal{OR}$ and $\mathcal{L}$. △: Significant differences ($p<0.05$) between $\mathcal{L}$ and $\mathcal{NL}$. §: Significant differences ($p<0.05$) between $\mathcal{NL}$ and $\mathcal{OR}$.

(a) | |||||
---|---|---|---|---|---|

Method | Metric | Deletion Probability (%) | |||

5 | 15 | 25 | 35 | ||

$\mathcal{OR}$ | MHR | 0.13 (0.05–0.24) | 0.25 (0.12–0.48) ${}^{\u2020}$ | 0.39 (0.20–0.74) ${}^{\u2020}$ | 0.54 (0.28–1.02) ${}^{\u2020}$ |

SDNN | 1.80 (0.85–3.07) ${}^{\u2020}$ | 3.61 (1.76–6.03) ${}^{\u2020}$ | 5.10 (2.34–9.18) ${}^{\u2020}$ | 7.32 (3.11–14.93) ${}^{\u2020}$ | |

RMSSD | 2.09 (0.95–4.03) ${}^{\u2020}$ | 5.40 (2.12–9.23) ${}^{\u2020}$ | 8.90 (3.96–14.55) | 10.84 (5.21–23.97) | |

$\mathcal{L}$ | MHR | 0.00 (0.00–0.01) ${}^{\u25b5}$ | 0.01 (0.01–0.03) ${}^{\u25b5}$ | 0.03 (0.01–0.49) ${}^{\u25b5}$ | 0.08 (0.03–0.75) ${}^{\u25b5}$ |

SDNN | 0.43 (0.19–0.81) ${}^{\u25b5}$ | 1.85 (0.81–3.37) ${}^{\u25b5}$ | 4.71 (2.59–8.25) ${}^{\u25b5}$ | 8.09 (4.18–14.48) ${}^{\u25b5}$ | |

RMSSD | 1.07 (0.41–2.05) ${}^{\u25b5}$ | 2.68 (1.14–6.09) ${}^{\u25b5}$ | 7.90 (2.72–19.56) ${}^{\u25b5}$ | 13.98 (5.21–37.84) ${}^{\u25b5}$ | |

$\mathcal{NL}$ | MHR | 0.00 (0.00–0.00) | 0.00 (0.00–0.02) | 0.02 (0.01–0.09) | 0.05 (0.01–0.70) ${}^{\mathrm{\S}}$ |

SDNN | 0.16 (0.05–0.44) ${}^{\mathrm{\S}}$ | 0.77 (0.24–2.34) ${}^{\mathrm{\S}}$ | 2.63 (0.85–6.00) ${}^{\mathrm{\S}}$ | 5.42 (2.10–10.18) ${}^{\mathrm{\S}}$ | |

RMSSD | 1.13 (0.44–2.28) ${}^{\mathrm{\S}}$ | 4.14 (2.13–7.43) ${}^{\mathrm{\S}}$ | 9.42 (4.95–16.70) ${}^{\mathrm{\S}}$ | 14.50 (7.54–29.35) ${}^{\mathrm{\S}}$ | |

(b) | |||||

Method | Metric | Burst duration (s) | |||

5 | 10 | 15 | 20 | ||

$\mathcal{OR}$ | MHR | 0.16 (0.07–0.28) | 0.22 (0.11–0.44) ${}^{\u2020}$ | 0.31 (0.14–0.56) ${}^{\u2020}$ | 0.40 (0.17–0.71) ${}^{\u2020}$ |

SDNN | 1.69 (0.84–2.43) ${}^{\u2020}$ | 2.12 (1.10–3.65) ${}^{\u2020}$ | 3.06 (1.40–4.87) ${}^{\u2020}$ | 3.55 (1.72–5.96) ${}^{\u2020}$ | |

RMSSD | 1.66 (0.83–2.44) ${}^{\u2020}$ | 2.43 (1.14–3.74) ${}^{\u2020}$ | 3.15 (1.58–5.38) ${}^{\u2020}$ | 4.08 (2.07–7.08) ${}^{\u2020}$ | |

$\mathcal{L}$ | MHR | 0.01 (0.00–0.03) ${}^{\u25b5}$ | 0.03 (0.01–0.53) ${}^{\u25b5}$ | 0.47 (0.02–0.76) ${}^{\u25b5}$ | 0.73 (0.09–0.98) ${}^{\u25b5}$ |

SDNN | 1.39 (0.57–2.66) ${}^{\u25b5}$ | 3.41 (1.51–5.32) ${}^{\u25b5}$ | 4.83 (2.63–7.82) ${}^{\u25b5}$ | 6.38 (3.14–9.87) ${}^{\u25b5}$ | |

RMSSD | 1.66 (0.75–3.38) ${}^{\u25b5}$ | 3.60 (1.83–6.23) ${}^{\u25b5}$ | 4.87 (2.84–8.41) ${}^{\u25b5}$ | 6.97 (3.95–10.86) ${}^{\u25b5}$ | |

$\mathcal{NL}$ | MHR | 0.01 (0.00–0.09) ${}^{\mathrm{\S}}$ | 0.02 (0.01–0.57) ${}^{\mathrm{\S}}$ | 0.09 (0.01–0.72) | 0.55 (0.03–0.83) |

SDNN | 1.24 (0.44–3.02) ${}^{\mathrm{\S}}$ | 2.84 (1.04–4.82) ${}^{\mathrm{\S}}$ | 4.46 (2.15–7.02) ${}^{\mathrm{\S}}$ | 5.80 (2.86–8.85) ${}^{\mathrm{\S}}$ | |

RMSSD | 1.77 (0.90–3.87) ${}^{\mathrm{\S}}$ | 3.77 (2.13–6.51) ${}^{\mathrm{\S}}$ | 5.80 (3.40–9.01) ${}^{\mathrm{\S}}$ | 7.78 (4.36–12.17) ${}^{\mathrm{\S}}$ |

**Table 2.**

**p**

**-values of ranked signed test for supine/tilt discrimination of time-domain metrics.**N.S.: Not significant ($p>0.05$).

Method | Metric | Reference | Deletion Probability (%) | Burst Duration (s) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

5 | 15 | 25 | 35 | 5 | 10 | 15 | 20 | |||||

$\mathcal{OR}$ | MHR | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

SDNN | 0.020 | 0.011 | 0.016 | N.S. | N.S. | 0.011 | 0.011 | 0.011 | 0.011 | |||

RMSSD | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

$\mathcal{L}$ | MHR | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

SDNN | 0.020 | 0.014 | 0.008 | N.S. | N.S. | 0.011 | 0.011 | 0.011 | 0.004 | |||

RMSSD | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

$\mathcal{NL}$ | MHR | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

SDNN | 0.020 | 0.012 | 0.011 | 0.021 | 0.014 | 0.014 | 0.007 | 0.012 | 0.026 | |||

RMSSD | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ |

**Table 3.**

**Relative error (%) of time-domain metrics from Apple Watch dataset.**†: Significant differences ($p<0.05$) between $\mathcal{OR}$ and $\mathcal{L}$. △: Significant differences ($p<0.05$) between $\mathcal{L}$ and $\mathcal{NL}$. §: Significant differences ($p<0.05$) between $\mathcal{NL}$ and $\mathcal{OR}$.

Metric | Method | ||
---|---|---|---|

$\mathcal{OR}$ | $\mathcal{L}$ | $\mathcal{NL}$ | |

MHR | 0.12 (0.04–0.47) | 0.03 (0.01–0.52) ${}^{\u25b5}$ | 0.03 (0.01–0.66) ${}^{\mathrm{\S}}$ |

SDNN | 3.36 (1.97–7.47) ${}^{\u2020}$ | 2.92 (1.51–9.55) ${}^{\u25b5}$ | 2.96 (1.31–8.62) |

RMSSD | 7.84 (4.29–15.90) ${}^{\u2020}$ | 8.56 (3.99–20.22) ${}^{\u25b5}$ | 8.61 (3.74–17.69) ${}^{\mathrm{\S}}$ |

**Table 4.**

**Relative error (%) of frequency-domain metrics computed via FFT. (a)**Scattered missing beats.

**(b)**Bursts. †: Significant differences ($p<0.05$) between $\mathcal{M}$ and $\mathcal{L}$. △: Significant differences ($p<0.05$) between $\mathcal{L}$ and $\mathcal{NL}$. §: Significant differences ($p<0.05$) between $\mathcal{NL}$ and $\mathcal{M}$.

(a) | |||||
---|---|---|---|---|---|

Method | Metric | Deletion Probability (%) | |||

5 | 15 | 25 | 35 | ||

$\mathcal{M}$ | ${P}_{LF}$ | 8.08 (3.46–19.10) ${}^{\u2020}$ | 20.29 (11.37–32.72) ${}^{\u2020}$ | 36.36 (22.36–53.64) ${}^{\u2020}$ | 55.16 (29.10–161.86) ${}^{\u2020}$ |

${P}_{HF}$ | 15.37 (7.00–30.10) ${}^{\u2020}$ | 32.89 (20.17–45.24) ${}^{\u2020}$ | 50.12 (36.46–63.16) ${}^{\u2020}$ | 59.41 (42.65–73.21) ${}^{\u2020}$ | |

$\mathcal{L}$ | ${P}_{LF}$ | 0.99 (0.39–2.34) ${}^{\u25b5}$ | 4.28 (2.04–9.24) ${}^{\u25b5}$ | 10.94 (5.66–18.71) ${}^{\u25b5}$ | 15.98 (8.81–28.74) ${}^{\u25b5}$ |

${P}_{HF}$ | 2.81 (1.19–5.47) ${}^{\u25b5}$ | 10.83 (5.54–17.64) ${}^{\u25b5}$ | 22.69 (11.98–41.40) ${}^{\u25b5}$ | 34.04 (19.54–61.20) | |

$\mathcal{NL}$ | ${P}_{LF}$ | 0.41 (0.15–1.11) ${}^{\mathrm{\S}}$ | 1.44 (0.48–4.71) ${}^{\mathrm{\S}}$ | 4.24 (1.41–12.57) ${}^{\mathrm{\S}}$ | 8.96 (2.20–21.22) ${}^{\mathrm{\S}}$ |

${P}_{HF}$ | 1.63 (0.71–4.16) ${}^{\mathrm{\S}}$ | 6.88 (2.45–14.99) ${}^{\mathrm{\S}}$ | 18.97 (9.80–37.55) ${}^{\mathrm{\S}}$ | 29.20 (17.06–54.77) ${}^{\mathrm{\S}}$ | |

(b) | |||||

Method | Metric | Burst duration (s) | |||

5 | 10 | 15 | 20 | ||

$\mathcal{M}$ | ${P}_{LF}$ | 10.20 (3.53–20.75) ${}^{\u2020}$ | 14.62 (6.52–26.29) ${}^{\u2020}$ | 21.90 (9.95–32.57) ${}^{\u2020}$ | 26.50 (14.26–39.04) ${}^{\u2020}$ |

${P}_{HF}$ | 12.62 (6.38–28.40) ${}^{\u2020}$ | 17.99 (9.32–32.90) ${}^{\u2020}$ | 22.82 (14.13–36.28) ${}^{\u2020}$ | 28.65 (18.53–43.37) ${}^{\u2020}$ | |

$\mathcal{L}$ | ${P}_{LF}$ | 4.94 (1.70–12.34) ${}^{\u25b5}$ | 10.89 (4.67–19.12) ${}^{\u25b5}$ | 15.45 (7.13–26.16) ${}^{\u25b5}$ | 19.25 (8.88–31.24) |

${P}_{HF}$ | 6.81 (2.92–11.42) ${}^{\u25b5}$ | 9.95 (4.98–17.20) ${}^{\u25b5}$ | 14.02 (7.06–22.67) ${}^{\u25b5}$ | 18.26 (9.41–28.11) ${}^{\u25b5}$ | |

$\mathcal{NL}$ | ${P}_{LF}$ | 4.72 (1.56–12.18) ${}^{\mathrm{\S}}$ | 10.01 (4.34–17.88) ${}^{\mathrm{\S}}$ | 13.31 (6.35–25.36) ${}^{\mathrm{\S}}$ | 19.34 (9.28–30.70) ${}^{\mathrm{\S}}$ |

${P}_{HF}$ | 6.82 (3.36–11.76) ${}^{\mathrm{\S}}$ | 11.02 (5.73–17.59) ${}^{\mathrm{\S}}$ | 15.19 (7.85–23.70) ${}^{\mathrm{\S}}$ | 19.10 (10.94–29.80) ${}^{\mathrm{\S}}$ |

**Table 5.**

**p**

**-values of ranked signed test for supine/tilt discrimination of frequency-domain metrics computed via FFT.**N.S.: Not significant ($p>0.05$).

Method | Metric | Reference | Deletion Probability (%) | Burst Duration (s) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

5 | 15 | 25 | 35 | 5 | 10 | 15 | 20 | |||||

$\mathcal{M}$ | ${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | N.S. | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

${P}_{LFn}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

${P}_{LF}$/${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | 0.005 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

$\mathcal{L}$ | ${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

${P}_{LFn}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

${P}_{LF}$/${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

$\mathcal{NL}$ | ${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

${P}_{LFn}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

${P}_{LF}$/${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ |

**Table 6.**

**Relative error (%) of frequency-domain metrics computed via FFT from Apple Watch dataset.**†: Significant differences ($p<0.05$) between $\mathcal{M}$ and $\mathcal{L}$. △: Significant differences ($p<0.05$) between $\mathcal{L}$ and $\mathcal{NL}$. §: Significant differences ($p<0.05$) between $\mathcal{NL}$ and $\mathcal{M}$.

Metric | Method | ||
---|---|---|---|

$\mathcal{M}$ | $\mathcal{L}$ | $\mathcal{NL}$ | |

${P}_{LF}$ | 0.09 (0.04–0.30) ${}^{\u2020}$ | 0.08 (0.03–0.22) ${}^{\u25b5}$ | 0.08 (0.03–0.17) ${}^{\mathrm{\S}}$ |

${P}_{HF}$ | 0.14 (0.07–0.31) ${}^{\u2020}$ | 0.16 (0.07–0.30) ${}^{\u25b5}$ | 0.17 (0.07–0.31) ${}^{\mathrm{\S}}$ |

**Table 7.**

**Relative error (%) of frequency-domain metrics computed via Lomb’s method. (a)**Scattered missing beats.

**(b)**Bursts. †: Significant differences ($p<0.05$) between $\mathcal{OR}$ and $\mathcal{L}$. △: Significant differences ($p<0.05$) between $\mathcal{L}$ and $\mathcal{NL}$. §: Significant differences ($p<0.05$) between $\mathcal{NL}$ and $\mathcal{OR}$.

(a) | |||||
---|---|---|---|---|---|

Method | Metric | Deletion Probability (%) | |||

5 | 15 | 25 | 35 | ||

$\mathcal{OR}$ | ${P}_{LF}$ | 10.75 (4.77–18.84) ${}^{\u2020}$ | 23.45 (9.90–40.99) ${}^{\u2020}$ | 34.71 (17.49–66.27) ${}^{\u2020}$ | 58.11 (24.93–123.23) ${}^{\u2020}$ |

${P}_{HF}$ | 23.01 (11.38–45.74) ${}^{\u2020}$ | 79.42 (46.93–155.29) ${}^{\u2020}$ | 160.28 (87.39–296.90) ${}^{\u2020}$ | 304.57 (142.89–665.16) ${}^{\u2020}$ | |

$\mathcal{L}$ | ${P}_{LF}$ | 0.89 (0.35–1.90) ${}^{\u25b5}$ | 3.75 (1.60–7.49) ${}^{\u25b5}$ | 9.90 (4.56–18.19) ${}^{\u25b5}$ | 15.77 (7.56–28.59) ${}^{\u25b5}$ |

${P}_{HF}$ | 2.62 (1.10–4.81) ${}^{\u25b5}$ | 8.94 (4.93–16.22) ${}^{\u25b5}$ | 21.27 (11.03–37.15) ${}^{\u25b5}$ | 30.78 (17.23–61.62) ${}^{\u25b5}$ | |

$\mathcal{NL}$ | ${P}_{LF}$ | 0.37 (0.13–1.06) ${}^{\mathrm{\S}}$ | 1.36 (0.44–3.95) ${}^{\mathrm{\S}}$ | 3.43 (1.17–11.70) ${}^{\mathrm{\S}}$ | 7.58 (2.28–22.67) ${}^{\mathrm{\S}}$ |

${P}_{HF}$ | 1.45 (0.51–3.31) ${}^{\mathrm{\S}}$ | 5.46 (1.92–12.01) ${}^{\mathrm{\S}}$ | 16.22 (8.12–31.57) ${}^{\mathrm{\S}}$ | 28.33 (14.72–52.65) ${}^{\mathrm{\S}}$ | |

(b) | |||||

Method | Metric | Burst Duration (s) | |||

5 | 10 | 15 | 20 | ||

$\mathcal{OR}$ | ${P}_{LF}$ | 11.19 (6.69–17.18) ${}^{\u2020}$ | 18.66 (10.87–28.82) ${}^{\u2020}$ | 25.33 (12.89–38.36) ${}^{\u2020}$ | 29.23 (14.57–48.91) ${}^{\u2020}$ |

${P}_{HF}$ | 14.06 (7.55–19.79) ${}^{\u2020}$ | 22.86 (12.70–34.06) ${}^{\u2020}$ | 30.88 (18.39–45.27) ${}^{\u2020}$ | 39.17 (24.01–60.79) ${}^{\u2020}$ | |

$\mathcal{L}$ | ${P}_{LF}$ | 4.48 (1.65–11.24) ${}^{\u25b5}$ | 9.99 (3.64–19.00) ${}^{\u25b5}$ | 13.85 (6.53–23.87) ${}^{\u25b5}$ | 17.38 (7.14–28.54) ${}^{\u25b5}$ |

${P}_{HF}$ | 5.51 (2.26–11.05) ${}^{\u25b5}$ | 8.74 (3.94–18.11) ${}^{\u25b5}$ | 13.18 (5.10–21.58) ${}^{\u25b5}$ | 16.22 (7.42–26.31) ${}^{\u25b5}$ | |

$\mathcal{NL}$ | ${P}_{LF}$ | 4.58 (1.68–11.53) ${}^{\mathrm{\S}}$ | 8.43 (3.55–17.21) ${}^{\mathrm{\S}}$ | 13.49 (5.93–23.56) ${}^{\mathrm{\S}}$ | 18.41 (9.23–28.86) ${}^{\mathrm{\S}}$ |

${P}_{HF}$ | 6.00 (2.79–11.69) ${}^{\mathrm{\S}}$ | 10.92 (5.24–19.42) ${}^{\mathrm{\S}}$ | 14.60 (7.65–23.30) ${}^{\mathrm{\S}}$ | 18.46 (9.45–29.31) ${}^{\mathrm{\S}}$ |

**Table 8.**

**p**

**-values of ranked signed test for supine/tilt discrimination of frequency-domain metrics computed via Lomb’s method.**N.S.: Not significant ($p>0.05$).

Method | Metric | Reference | Deletion Probability (%) | Burst Duration (s) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

5 | 15 | 25 | 35 | 5 | 10 | 15 | 20 | |||||

$\mathcal{OR}$ | ${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

${P}_{LFn}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

${P}_{LF}$/${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

$\mathcal{L}$ | ${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

${P}_{LFn}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

${P}_{LF}$/${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

$\mathcal{NL}$ | ${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

${P}_{LFn}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

${P}_{LF}$/${P}_{HF}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ |

**Table 9.**

**Relative error (%) of frequency-domain computed via Lomb’s method metrics from Apple Watch dataset.**†: Significant differences ($p<0.05$) between $\mathcal{OR}$ and $\mathcal{L}$. △: Significant differences ($p<0.05$) between $\mathcal{L}$ and $\mathcal{NL}$. §: Significant differences ($p<0.05$) between $\mathcal{NL}$ and $\mathcal{OR}$.

Metric | Method | ||
---|---|---|---|

$\mathcal{OR}$ | $\mathcal{L}$ | $\mathcal{NL}$ | |

${P}_{LF}$ | 0.10 (0.05–0.24) ${}^{\u2020}$ | 0.08 (0.03–0.23) ${}^{\u25b5}$ | 0.08 (0.03–0.18) ${}^{\mathrm{\S}}$ |

${P}_{HF}$ | 0.20 (0.08–0.56) ${}^{\u2020}$ | 0.15 (0.06–0.32) ${}^{\u25b5}$ | 0.13 (0.06–0.25) ${}^{\mathrm{\S}}$ |

**Table 10.**

**Relative error (%) of Poincaré metrics. (a)**Scattered missing beats.

**(b)**Bursts. †: Significant differences ($p<0.05$) between $\mathcal{OR}$ and $\mathcal{L}$. △: Significant differences ($p<0.05$) between $\mathcal{L}$ and $\mathcal{NL}$. §: Significant differences ($p<0.05$) between $\mathcal{NL}$ and $\mathcal{OR}$.

(a) | |||||
---|---|---|---|---|---|

Method | Metric | Deletion Probability (%) | |||

5 | 15 | 25 | 35 | ||

$\mathcal{OR}$ | SD1 | 2.13 (0.94–3.96) ${}^{\u2020}$ | 5.33 (2.24–9.25) ${}^{\u2020}$ | 9.06 (4.10–14.36) | 10.66 (5.29–23.67) |

SD2 | 2.58 (1.31–4.28) ${}^{\u2020}$ | 4.93 (2.04–8.53) ${}^{\u2020}$ | 7.20 (3.22–13.05) ${}^{\u2020}$ | 10.75 (4.93–20.32) ${}^{\u2020}$ | |

Md | 2.40 (1.16–4.15) ${}^{\u2020}$ | 4.49 (2.18–7.44) ${}^{\u2020}$ | 6.35 (2.87–10.78) ${}^{\u2020}$ | 8.70 (4.10–17.82) ${}^{\u2020}$ | |

Sd | 2.80 (1.31–4.95) ${}^{\u2020}$ | 6.19 (2.87–11.14) | 10.05 (4.47–18.67) ${}^{\u2020}$ | 14.46 (7.16–31.01) ${}^{\u2020}$ | |

$\mathcal{L}$ | SD1 | 1.07 (0.41–2.05) ${}^{\u25b5}$ | 2.68 (1.14–6.09) ${}^{\u25b5}$ | 7.90 (2.72–19.56) ${}^{\u25b5}$ | 14.00 (5.21–37.87) ${}^{\u25b5}$ |

SD2 | 0.40 (0.19–0.93) ${}^{\u25b5}$ | 1.85 (0.97–3.30) ${}^{\u25b5}$ | 4.28 (2.24–7.08) ${}^{\u25b5}$ | 6.99 (3.95–13.22) ${}^{\u25b5}$ | |

Md | 0.52 (0.22–1.08) ${}^{\u25b5}$ | 2.05 (0.85–4.18) ${}^{\u25b5}$ | 4.46 (2.07–7.23) ${}^{\u25b5}$ | 7.11 (3.54–11.93) ${}^{\u25b5}$ | |

Sd | 0.47 (0.17–0.98) ${}^{\u25b5}$ | 1.34 (0.52–3.23) ${}^{\u25b5}$ | 3.59 (1.27–12.57) ${}^{\u25b5}$ | 7.09 (1.78–33.63) ${}^{\u25b5}$ | |

$\mathcal{NL}$ | SD1 | 1.13 (0.44–2.28) ${}^{\mathrm{\S}}$ | 4.14 (2.13–7.43) ${}^{\mathrm{\S}}$ | 9.42 (4.95–16.70) ${}^{\mathrm{\S}}$ | 14.51 (7.54–29.39) ${}^{\mathrm{\S}}$ |

SD2 | 0.12 (0.04–0.30) ${}^{\mathrm{\S}}$ | 0.56 (0.15–1.50) ${}^{\mathrm{\S}}$ | 1.67 (0.54–3.98) ${}^{\mathrm{\S}}$ | 3.39 (1.19–8.51) ${}^{\mathrm{\S}}$ | |

Md | 0.23 (0.09–0.56) ${}^{\mathrm{\S}}$ | 1.06 (0.27–3.00) ${}^{\mathrm{\S}}$ | 2.43 (1.04–5.73) ${}^{\mathrm{\S}}$ | 4.83 (1.88–9.33) ${}^{\mathrm{\S}}$ | |

Sd | 0.30 (0.10–0.76) ${}^{\mathrm{\S}}$ | 0.90 (0.26–2.46) ${}^{\mathrm{\S}}$ | 2.39 (0.77–7.89) ${}^{\mathrm{\S}}$ | 4.92 (1.60–17.24) ${}^{\mathrm{\S}}$ | |

Method | Metric | Burst Duration (s) | |||

5 | 10 | 15 | 20 | ||

$\mathcal{OR}$ | SD1 | 1.69 (0.84–2.50) ${}^{\u2020}$ | 2.45 (1.16–3.76) ${}^{\u2020}$ | 3.19 (1.63–5.39) ${}^{\u2020}$ | 4.03 (2.00–7.04) ${}^{\u2020}$ |

SD2 | 1.85 (0.89–2.74) ${}^{\u2020}$ | 2.44 (1.29–3.92) ${}^{\u2020}$ | 3.24 (1.57–5.11) ${}^{\u2020}$ | 3.94 (1.89–6.39) ${}^{\u2020}$ | |

Md | 1.91 (0.94–3.05) ${}^{\u2020}$ | 2.54 (1.04–4.59) ${}^{\u2020}$ | 3.49 (1.26–5.86) ${}^{\u2020}$ | 4.14 (2.01–7.40) ${}^{\u2020}$ | |

Sd | 1.50 (0.90–2.49) | 2.32 (1.21–3.65) | 2.98 (1.65–4.54) | 3.57 (2.02–5.67)${}^{\u2020}$ | |

$\mathcal{L}$ | SD1 | 1.67 (0.75–3.38) ${}^{\u25b5}$ | 3.60 (1.83–6.23) ${}^{\u25b5}$ | 4.88 (2.84–8.41) ${}^{\u25b5}$ | 6.97 (3.95–10.86) ${}^{\u25b5}$ |

SD2 | 1.31 (0.56–2.70) ${}^{\u25b5}$ | 3.40 (1.36–5.37) ${}^{\u25b5}$ | 4.90 (2.49–7.88) ${}^{\u25b5}$ | 6.26 (3.10–10.14) ${}^{\u25b5}$ | |

Md | 2.33 (0.97–4.11) ${}^{\u25b5}$ | 5.65 (2.90–8.72) ${}^{\u25b5}$ | 8.44 (4.31–11.97) ${}^{\u25b5}$ | 10.53 (4.53–14.29) ${}^{\u25b5}$ | |

Sd | 1.97 (0.75–4.06) ${}^{\u25b5}$ | 3.24 (1.57–6.84) | 4.08 (1.92–7.50) | 4.68 (2.37–8.05) ${}^{\u25b5}$ | |

$\mathcal{NL}$ | SD1 | 1.77 (0.90–3.87) ${}^{\mathrm{\S}}$ | 3.78 (2.13–6.52) ${}^{\mathrm{\S}}$ | 5.80 (3.41–9.02) ${}^{\mathrm{\S}}$ | 7.78 (4.36–12.17) ${}^{\mathrm{\S}}$ |

SD2 | 1.16 (0.32–2.87) ${}^{\mathrm{\S}}$ | 2.53 (0.89–4.96) ${}^{\mathrm{\S}}$ | 4.42 (2.18–6.88) ${}^{\mathrm{\S}}$ | 5.60 (2.61–8.70) ${}^{\mathrm{\S}}$ | |

Md | 1.16 (0.32–2.87) ${}^{\mathrm{\S}}$ | 2.53 (0.89–4.96) ${}^{\mathrm{\S}}$ | 4.42 (2.18–6.88) ${}^{\mathrm{\S}}$ | 5.60 (2.61–8.70) ${}^{\mathrm{\S}}$ | |

Sd | 1.81 (0.61–4.33) | 2.97 (1.21–5.62) | 3.53 (1.67–7.63) | 4.17 (1.99–7.03) |

**Table 11.**

**p**

**-values of ranked signed test for supine/tilt discrimination of Poincaré metrics.**N.S.: Not Significative ($p>0.05$).

Method | Metric | Reference | Deletion Probability (%) | Burst Duration (s) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

5 | 15 | 25 | 35 | 5 | 10 | 15 | 20 | |||||

$\mathcal{OR}$ | SD1 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

SD2 | 0.031 | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | |||

SD12 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

S | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | 0.012 | 0.002 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

Md | <${10}^{-3}$ | 0.001 | 0.002 | 0.155 | 0.016 | 0.001 | 0.002 | 0.002 | 0.002 | |||

Sd | 0.039 | 0.009 | N.S. | N.S. | N.S. | 0.024 | 0.026 | 0.022 | 0.015 | |||

$\mathcal{L}$ | SD1 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

SD2 | 0.031 | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | |||

SD12 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

S | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | 0.007 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

Md | <${10}^{-3}$ | <${10}^{-3}$ | 0.001 | 0.010 | 0.012 | <${10}^{-3}$ | <${10}^{-3}$ | 0.002 | 0.002 | |||

Sd | 0.039 | 0.034 | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | |||

$\mathcal{NL}$ | SD1 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | ||

SD2 | 0.031 | 0.043 | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | N.S. | |||

SD12 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

S | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | 0.004 | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | |||

Md | <${10}^{-3}$ | <${10}^{-3}$ | <${10}^{-3}$ | 0.002 | 0.002 | 0.002 | 0.001 | 0.002 | 0.013 | |||

Sd | 0.039 | 0.033 | N.S. | N.S. | N.S. | 0.041 | 0.041 | 0.027 | 0.049 |

**Table 12.**

**Relative error (%) of Poincaré metrics from Apple Watch dataset.**†: Significant differences ($p<0.05$) between $\mathcal{OR}$ and $\mathcal{L}$. △: Significant differences ($p<0.05$) between $\mathcal{L}$ and $\mathcal{NL}$. §: Significant differences ($p<0.05$) between $\mathcal{NL}$ and $\mathcal{OR}$.

Metric | Method | ||
---|---|---|---|

$\mathcal{OR}$ | $\mathcal{L}$ | $\mathcal{NL}$ | |

SD1 | 7.83 (4.15–15.91) ${}^{\u2020}$ | 8.56 (3.99–20.22) ${}^{\u25b5}$ | 8.61 (3.73–17.70) ${}^{\mathrm{\S}}$ |

SD2 | 3.22 (1.59–6.38) ${}^{\u2020}$ | 2.61 (1.06–7.98) ${}^{\u25b5}$ | 2.35 (1.06–6.08) |

Md | 3.97 (2.21–8.28) ${}^{\u2020}$ | 3.55 (2.03–9.24) ${}^{\u25b5}$ | 3.40 (1.67–8.42) |

Sd | 4.20 (1.66–9.42) | 3.96 (1.87–10.00) ${}^{\u25b5}$ | 3.44 (1.49–10.06) ${}^{\mathrm{\S}}$ |

**Table 13.**Summary of findings.

**(a)**Best correction method.

**(b)**Maximum acceptable missing beats for a relative error less than 20% in the third quartile.

(a) | ||
---|---|---|

Metric | Scattered Missing Beats | Bursts |

MHR | $\mathcal{NL}$ | $\mathcal{NL}$ |

SDNN | $\mathcal{NL}$ | $\mathcal{OR}$ |

RMSSD | $\mathcal{L}$ | $\mathcal{OR}$ |

${P}_{LF}$ (FFT) | $\mathcal{NL}$ | $\mathcal{NL}$ |

${P}_{HF}$ (FFT) | $\mathcal{NL}$ | $\mathcal{L}$ |

${P}_{LF}$ (Lomb) | $\mathcal{NL}$ | $\mathcal{NL}$ |

${P}_{HF}$ (Lomb) | $\mathcal{NL}$ | $\mathcal{L}$ |

SD1 | $\mathcal{L}$ | $\mathcal{OR}$ |

SD2 | $\mathcal{NL}$ | $\mathcal{OR}$ |

Md | $\mathcal{NL}$ | $\mathcal{OR}$ |

Sd | $\mathcal{NL}$ | $\mathcal{OR}$/$\mathcal{NL}$ |

(b) | ||

Metric | Scattered Missing Beats | Bursts |

MHR | 35% | 20 s |

SDNN | 35% | 20 s |

RMSSD | 25% | 20 s |

${P}_{LF}$ (FFT) | 25% | 10 s |

${P}_{HF}$ (FFT) | 15% | 10 s |

${P}_{LF}$ (Lomb) | 25% | 10 s |

${P}_{HF}$ (Lomb) | 15% | 10 s |

SD1 | 25% | 20 s |

SD2 | 35% | 20 s |

Md | 35% | 20 s |

Sd | 35% | 20 s |

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## Share and Cite

**MDPI and ACS Style**

Cajal, D.; Hernando, D.; Lázaro, J.; Laguna, P.; Gil, E.; Bailón, R.
Effects of Missing Data on Heart Rate Variability Metrics. *Sensors* **2022**, *22*, 5774.
https://doi.org/10.3390/s22155774

**AMA Style**

Cajal D, Hernando D, Lázaro J, Laguna P, Gil E, Bailón R.
Effects of Missing Data on Heart Rate Variability Metrics. *Sensors*. 2022; 22(15):5774.
https://doi.org/10.3390/s22155774

**Chicago/Turabian Style**

Cajal, Diego, David Hernando, Jesús Lázaro, Pablo Laguna, Eduardo Gil, and Raquel Bailón.
2022. "Effects of Missing Data on Heart Rate Variability Metrics" *Sensors* 22, no. 15: 5774.
https://doi.org/10.3390/s22155774