# Analysis of the Impact of Interpolation Methods of Missing RR-intervals Caused by Motion Artifacts on HRV Features Estimations

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## Abstract

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## 1. Introduction

#### 1.1. Paper Contribution

#### 1.2. Related Work

## 2. Materials and Methods

#### 2.1. Dataset

#### 2.2. Missing Values Interpolation

- No interpolation: this approach does not create interpolated values of missing RR-intervals. Differently to the Deletion method that remove missing values merging non-consecutive beats that induce in missing interpretation of HRV features, the no-interpolation method maintains the missing values into the RR-intervals time series.
- Nearest neighbor: the nearest neighbor or proximate interpolation is the easiest interpolation method [12]. This interpolation assigns the value of the closest known (existing) neighbor to the missing- value as shows in Equation (6).$${X}_{i}=\left\{\begin{array}{cc}{x}_{B}\hfill & \mathrm{if}i\frac{a+b}{2}\hfill \\ {x}_{A}\hfill & \mathrm{if}i\ge \frac{a+b}{2}\hfill \end{array}\right.$$
- Linear: this method fits a straight line passing through points ${x}_{A}$ and ${x}_{B}$ [14]. Interpolated data by the linear model are bound between ${x}_{A}$ and ${x}_{B}$ as showed in Equation (7).$${X}_{i}=\frac{{x}_{A}-{x}_{B}}{a-b}(i-b)+{x}_{B}.$$
- Quadratic: differently from the linear interpolation model, the quadratic function needs three points of interest to interpolate missing values in a time series as showed in Equation (8).$${X}_{i}={x}_{B}\frac{(i-b)({x}_{C}-{x}_{A})}{2(b-a)}+\frac{{(i-b)}^{2}({x}_{A}-2{x}_{B}+{x}_{C})}{2{(b-a)}^{2}}.$$
- Spline cubic: fitting datapoints using polynomials of degree higher than one leads to problems of oscillation outside the fitted points, known as Runge’s phenomenon [15]. This problem can be avoided by using a spline, a function defined piecewise by polynomials, using datapoints as control points instead of forcing the fitted function to pass through the data points. Cubic spline is a spline composed of piecewise third-order polynomials. By using third degree polynomials is possible to ensure that the resulting curve is smooth [15], avoiding the problem of the straight polynomial interpolation that tends to induce distortions on the edges of the polynomials, given by the fact that, in general, the first and second derivative of the function defined by piecewise polynomials will not be continuous at the edges of polynomials. With cubic spline, it is possible to force the first and second derivatives of consecutive polynomials to be equal, ensuring smoothness of the resulting curve.

#### 2.3. Feature Engineering

- Time domain:
- -
- HR mean: mean values of heart rate (HR) computed as showed in Equation (9).$$H{R}_{mean}=\frac{1}{N-1}\sum _{i=1}^{N-1}60/({R}_{i+1}-{R}_{i}),$$
- -
- RMSSD: root mean square of the successive RR-intervals differences (Equation (10)) represents the strength of the autonomic nervous system (specifically the parasympathetic branch) at a given time.$$RMSSD=\sqrt{\frac{1}{N-1}\sum _{i=1}^{N-1}{[({R}_{i+1}-{R}_{i})-({R}_{i}-{R}_{i-1})]}^{2}},$$
- -
- SDNN: standard deviation of RR-intervals (Equation (11)). It reflects the cyclic components responsible for variability in the RR-intervals time series. The SDNN is the “gold standard” for medical stratification of both morbidity and mortality [16].$$SDNN=\sqrt{\frac{1}{N-1}\sum _{i=1}^{N}{(R{R}_{i}-\overline{RR})}^{2}},$$
- -
- PNN50: the ratio between NN50 (i.e., number of pairs of successive RR intervals that differ by more than 50 ms) and the total number of RR-intervals (Equation (12)).$$PNN50=\frac{NN{50}_{count}}{{N}_{RR-intervals}}$$

- Frequency domain:
- -
- Power spectral density (PSD): describes the distribution of power into frequency components composing that signal. The Lomb–Scargle periodogram for PSD estimation was found to be the most appropriate method to analyze RR-interval data [5,6]. VLF (power in very-low-frequency ranges, i.e., ≤0.04 Hz), LF (power in low-frequency ranges, i.e., 0.04–0.15 Hz), HF (Power in high-frequency ranges, i.e., 0.15, 0.4 Hz), LF/HF ratio (ratio between LF and HF expressed as ms${}^{2}$), and total power (Power in all the frequency ranges, i.e., ≤0.4) were obtained by the sum of the power in the relevant frequency range in the spectrum.

- Non-linear HRV features:
- -
- Poincaré plot: it is a type of recurrence plot used to quantify self-similarity in processes. A Poincaré plot is a graph of RR interval ($R{R}_{n}$) against the previous one ($R{R}_{n}-1$). From this scatter plot, it is possible to quantitatively analyze the variance of two consecutive RR-intervals by fitting an ellipse to the plotted shape. $SD1$ is the standard deviation of Poincaré plot perpendicular to the line-of-identity, while $SD2$ is the standard deviation of the Poincaré plot along the line-of-identity.

#### 2.4. Success Metrics

## 3. Results and Discussions

#### 3.1. Results Summary

#### 3.2. HRV Features

#### 3.2.1. Time Domain

#### 3.2.2. Frequency Domain

#### 3.2.3. Non-Linear Domain

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Examples of 30%, 50% and 70% of missing values created by Gilbert model. The colored lines refer to missing beats.

**Figure 4.**Difference between linear interpolation on time and duration. Red solid line refers to real variation of the timing between beats (RR-intervals) time series, green dashed line refers to the on-duration approach, and green dot dash line refers to the on-time approach.

**Figure 5.**Frequency analysis of a user’s RR-intervals timeseries recorded in 5 min with different percentages of missing values (i.e., 0%, 30%, 50% and 70%) handled with different interpolation methods (i.e., nearest neighbor, linear, quadratic and cubic spline) on both time and duration.

**Figure 6.**Poincaré plot of a user’s RR-intervals timeseries recorded in 5 min with different percentage of missing values (i.e., 0%, 30%, 50% and 70%) handled with different interpolation methods (i.e., nearest neighbor, linear, quadratic and cubic spline) on both time and duration.

**Table 1.**Difference between duration and time interpolation by using different approach (i.e., no-missing values, nearest neighbor, linear, quadratic, and cubic spline).

Window Time (s) | RMSE (s) | RE (%) | ||||
---|---|---|---|---|---|---|

Interpolation | Time | Duration | Time | Duration | Time | Duration |

No-missing values | 90.11 | — | — | |||

Nearest | — | 91.95 | — | 0.096 | — | 5.11 |

Linear | 90.11 | 91.83 | 0.075 | 0.090 | 3.70 | 4.86 |

Quadratic | 90.11 | 92.13 | 0.084 | 0.107 | 4.35 | 5.83 |

Cubic spline | 90.11 | 92.24 | 0.085 | 0.109 | 3.46 | 6.63 |

**Table 2.**Descriptive statistic of hart rate variability (HRV) features. Mean and 95% coefficient intervals (CI) are provided for all the feature.

HRV Features | Mean | 95% CI |
---|---|---|

IBI (s) | 0.78 | [0.54, 1.11] |

PNN50 (n) | 8 | [4, 16] |

RMSSD (s) | 0.039 | [0.017, 0.36] |

SD1 (s) | 0.027 | [0.012, 0.26] |

SD2 (s) | 0.077 | [0.040, 0.25] |

SDNN (s) | 0.059 | [0.017, 0.25] |

VLF (s${}^{2}$) | 0.87 | [0.22, 4.15] |

LF (s${}^{2}$) | 0.477 | [0.12, 5.57] |

HF (s${}^{2}$) | 0.28 | [0.050, 3.024] |

total power (s${}^{2}$) | 1.91 | [0.53, 21.44] |

LF/HF (s${}^{2}$) | 2.9 | [1.2, 10.2] |

**Table 3.**Best performing interpolation approach (i.e., with low RE) for each HRV feature in each percentage of missing values evaluated. The error in estimating HRV features is reported using RE and root mean squared error (RMSE).

Interpolation | |||||
---|---|---|---|---|---|

Missing Values (%) | HRV | How | Method | RE (%) | RMSE |

30 | RMSSD (s) | No-interpolation | 14.65 | 0.38 | |

SDNN (s) | Time | quadratic | 9.42 | 0.34 | |

PNN50 (n) | No-interpolation | 24.37 | 1.51 | ||

SD1 (s) | No-interpolation | 14.68 | 0.27 | ||

SD2 (s) | Time | quadratic | 8.57 | 0.47 | |

VLF (s${}^{2}$) | Time | quadratic | 14.50 | 0.82 | |

LF (s${}^{2}$) | Time | quadratic | 26.87 | 2.01 | |

HF (s${}^{2}$) | Time | quadratic | 32.18 | 4.48 | |

LF/HF (s${}^{2}$) | Time | cubic | 41.39 | 1.73 | |

total power (s${}^{2}$) | Time | quadratic | 17.16 | 6.26 | |

50 | RMSSD (ms) | No-interpolation | 23.13 | 0.76 | |

SDNN (s) | Time | quadratic | 15.47 | 0.41 | |

PNN50 (n) | No-interpolation | 39.01 | 2.35 | ||

SD1 (s) | No-interpolation | 23.18 | 0.54 | ||

SD2 (s) | Time | quadratic | 13.49 | 0.49 | |

VLF (s${}^{2}$) | Time | quadratic | 23.72 | 0.40 | |

LF (s${}^{2}$) | Time | quadratic | 42.42 | 1.12 | |

HF (s${}^{2}$) | Time | quadratic | 52.56 | 2.48 | |

LF/HF (s${}^{2}$) | Time | cubic | 58.07 | 2.26 | |

total power (s${}^{2}$) | Time | quadratic | 27.59 | 3.96 | |

70 | RMSSD (s) | No-interpolation | 34.37 | 0.91 | |

SDNN (s) | Time | quadratic | 22.76 | 0.47 | |

PNN50 (n) | Time | linear | 63.90 | 3.88 | |

SD1 (s) | No-interpolation | 34.46 | 0.59 | ||

SD2 (s) | Time | quadratic | 19.19 | 0.51 | |

VLF (s${}^{2}$) | Time | quadratic | 29.73 | 0.52 | |

LF (s${}^{2}$) | Time | quadratic | 56.41 | 1.45 | |

HF (s${}^{2}$) | Time | quadratic | 72.98 | 3.34 | |

LF/HF (s${}^{2}$) | Time | cubic | 72.07 | 2.80 | |

total power (s${}^{2}$) | Time | quadratic | 72.07 | 5.27 |

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**MDPI and ACS Style**

Morelli, D.; Rossi, A.; Cairo, M.; Clifton, D.A. Analysis of the Impact of Interpolation Methods of Missing RR-intervals Caused by Motion Artifacts on HRV Features Estimations. *Sensors* **2019**, *19*, 3163.
https://doi.org/10.3390/s19143163

**AMA Style**

Morelli D, Rossi A, Cairo M, Clifton DA. Analysis of the Impact of Interpolation Methods of Missing RR-intervals Caused by Motion Artifacts on HRV Features Estimations. *Sensors*. 2019; 19(14):3163.
https://doi.org/10.3390/s19143163

**Chicago/Turabian Style**

Morelli, Davide, Alessio Rossi, Massimo Cairo, and David A. Clifton. 2019. "Analysis of the Impact of Interpolation Methods of Missing RR-intervals Caused by Motion Artifacts on HRV Features Estimations" *Sensors* 19, no. 14: 3163.
https://doi.org/10.3390/s19143163