# Multiscale Distribution Entropy Analysis of Short-Term Heart Rate Variability

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sample Entropy

#### 2.2. Distribution Entropy

#### 2.3. Multiscale Distribution Entropy

#### 2.4. Evaluation Data

#### 2.4.1. Synthetic Data

#### 2.4.2. Real ECG Data

## 3. Results

#### 3.1. Simulation Result Using Synthetic Data

#### 3.2. Experiment Results Using Real Data

#### 3.2.1. ECG Dataset I

#### 3.2.2. ECG Dataset II

#### 3.2.3. Statistical Analysis for CHF Patients, Healthy Elderly, and Healthy Young Groups

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Empirical probability density function (ePDF) of white noise with length $N=1000$ ($m=2,r=0.15\sigma $).

**Figure 3.**Examples of inter-beat (RR) interval time series extracted from the electrocardiogram (ECG) signals of Dataset I: Congestive heart failure (CHF) patient (

**top panel**); Healthy elderly subject (

**middle panel**); Healthy young subject (

**bottom panel**).

**Figure 4.**Entropy values for synthetic data: (

**a**) multiscale entropy (MSE) value for N = 100; (

**b**) multiscale distribution entropy (MDE) value for $N=100$; (

**c**) MSE value for $N=300$; (

**d**) MDE value for $N=300$; (

**e**) MSE value for $N=1000$; (

**f**) MDE value for $N=1000$. Scales between 1 and 20 are used and the value at each scale represents a mean ± standard deviation.

**Figure 5.**MSE, multiscale permutation entropy (MPE), and MDE results using real data (RR interval) for CHF patients, healthy elderly and healthy young groups: (

**a**) MSE for $N=100$; (

**b**) MPE for $N=100$; (

**c**) MDE for $N=100$; (

**d**) MSE for $N=1000$; (

**e**) MPE for $N=1000$; (

**f**) MDE for $N=1000$. Scale range of 1–20 were used and the value at each scale represents a mean ± standard deviation.

**Figure 6.**MSE, MPE, and MDE results using real data (RR interval) for other CHF patients and normal sinus rhythm (NSR) subjects: (

**a**) MSE for $N=100$; (

**b**) MPE for $N=100$; (

**c**) MDE for $N=100$; (

**d**) MSE for $N=1000$; (

**e**) MPE for $N=1000$; (

**f**) MDE for $N=1000$. Scale range of 1–20 were used and the value at each scale represents a mean ± standard deviation.

**Table 1.**Statistical analysis results of MSE for RR interval of the CHF patients, healthy young, and healthy elder groups. The shadows mean that the distinction between the groups is unclear. C, E, and Y represent CHF, Elder, and Young, respectively. s denotes scale factor and N/A denotes ‘Not Available’.

MSE Statistical Results for RR Interval Time Series | |||||||||
---|---|---|---|---|---|---|---|---|---|

s | p-Value | ||||||||

$\mathit{N}=100$ | $\mathit{N}=300$ | $\mathit{N}=1000$ | |||||||

C - E | C - Y | E - Y | C - E | C - Y | E - Y | C - E | C - Y | E - Y | |

1 | 0.098 | 1.1 × 10^{−6} | 8.7 × 10^{−5} | 0.0014 | 2.5 × 10^{−8} | 3.6 × 10^{−4} | 1.3 × 10^{−5} | 4.8 × 10^{−14} | 8.5 × 10^{−7} |

2 | N/A | N/A | N/A | 4.7 × 10^{−7} | 5.5 × 10^{−13} | 1.0 × 10^{−5} | 6.7 × 10^{−10} | 1.5 × 10^{−21} | 8.0 × 10^{−15} |

3 | N/A | N/A | N/A | 2.4 × 10^{−10} | 1.8 × 10^{−17} | 1.9 × 10^{−4} | 8.4 × 10^{−14} | 3.2 × 10^{−25} | 5.3 × 10^{−10} |

4 | N/A | N/A | N/A | 3.9 × 10^{−14} | 3.1 × 10^{−19} | 0.019 | 2.3 × 10^{−17} | 2.2 × 10^{−26} | 2.7 × 10^{−07} |

5 | N/A | N/A | N/A | 1.3 × 10^{−11} | 2.8 × 10^{−15} | 0.311 | 1.3 × 10^{−20} | 1.9 × 10^{−28} | 0.005 |

6 | N/A | N/A | N/A | N/A | N/A | N/A | 1.9 × 10^{−21} | 8.8 × 10^{−27} | 0.089 |

7 | N/A | N/A | N/A | N/A | N/A | N/A | 3.7 × 10^{−19} | 1.2 × 10^{−22} | 0.512 |

8 | N/A | N/A | N/A | N/A | N/A | N/A | 1.6 × 10^{−20} | 1.1 × 10^{−22} | 0.847 |

9 | N/A | N/A | N/A | N/A | N/A | N/A | 3.5 × 10^{−21} | 2.2 × 10^{−23} | 0.353 |

10 | N/A | N/A | N/A | N/A | N/A | N/A | 1.4 × 10^{−19} | 5.1 × 10^{−20} | 0.175 |

11 | N/A | N/A | N/A | N/A | N/A | N/A | 5.2 × 10^{−18} | 6.8 × 10^{−20} | 0.673 |

12 | N/A | N/A | N/A | N/A | N/A | N/A | 1.6 × 10^{−19} | 2.3 × 10^{−21} | 0.359 |

13 | N/A | N/A | N/A | N/A | N/A | N/A | 1.3 × 10^{−17} | 1.0 × 10^{−17} | 0.086 |

14 | N/A | N/A | N/A | N/A | N/A | N/A | 3.7 × 10^{−18} | 5.0 × 10^{−20} | 0.853 |

15 | N/A | N/A | N/A | N/A | N/A | N/A | 1.9 × 10^{−13} | 2.2 × 10^{−14} | 0.448 |

16 | N/A | N/A | N/A | N/A | N/A | N/A | 1.6 × 10^{−13} | 4.7 × 10^{−16} | 0.945 |

17 | N/A | N/A | N/A | N/A | N/A | N/A | 1.0 × 10^{−11} | 1.6 × 10^{−14} | 0.695 |

18 | N/A | N/A | N/A | N/A | N/A | N/A | 1.6 × 10^{−15} | 2.2 × 10^{−19} | 0.936 |

19 | N/A | N/A | N/A | N/A | N/A | N/A | N/A | 3.2 ×10^{−16} | N/A |

20 | N/A | N/A | N/A | N/A | N/A | N/A | N/A | 4.1 × 10^{−13} | N/A |

**Table 2.**Statistical analysis results of MPE for RR interval of the CHF patients, healthy young, and healthy elder groups. The shadows mean that the distinction between the groups is unclear. C, E, and Y represent CHF, Elder, and Young, respectively. s denotes scale factor.

MPE Statistical Results for RR Interval Time Series | |||||||||
---|---|---|---|---|---|---|---|---|---|

s | p-Value | ||||||||

$\mathit{N}=100\text{}$ | $\mathit{N}=300$ | $\mathit{N}=1000$ | |||||||

C - E | C - Y | E - Y | C - E | C - Y | E - Y | C - E | C - Y | E - Y | |

1 | 0.389 | 6.6 × 10^{−7} | 6.5 × 10^{−5} | 0.255 | 1.2 × 10^{−7} | 8.9 × 10^{−6} | 0.157 | 8.6 × 10^{−8} | 5.6 × 10^{−4} |

2 | 0.018 | 0.101 | 5.8 × 10^{−5} | 0.006 | 0.006 | 1.1 × 10^{−7} | 0.005 | 4.1 × 10^{−4} | 7.9 × 10^{−7} |

3 | 3.0 × 10^{−4} | 0.519 | 2.9 × 10^{−4} | 4.6 × 10^{−6} | 0.005 | 1.6 × 10^{−14} | 9.2 × 10^{−9} | 0.012 | 5.9 × 10^{−13} |

4 | 0.363 | 1.9 × 10−^{5} | 1.8 × 10^{−4} | 0.674 | 8.6 × 10^{−13} | 1.5 × 10^{−16} | 0.030 | 5.2 × 10^{−12} | 4.8 × 10^{−19} |

5 | 0.139 | 6.3 × 10^{−8} | 0.001 | 0.935 | 1.6 × 10^{−9} | 7.3 × 10^{−13} | 0.776 | 1.8 × 10^{−15} | 1.2 × 10^{−16} |

6 | 0.059 | 0.004 | 0.241 | 0.033 | 4.2 × 10^{−9} | 1.1 × 10^{−5} | 0.017 | 7.5 × 10^{−16} | 9.5 × 10^{−10} |

7 | 0.102 | 1.2 × 10^{−8} | 0.655 | 0.002 | 3.7 × 10^{−7} | 0.007 | 0.007 | 8.8 × 10^{−13} | 3.2 × 10^{−6} |

8 | 0.292 | 5.8 × 10^{−7} | 0.619 | 0.004 | 4.9 × 10^{−4} | 0.428 | 1.1 × 10^{−5} | 3.5 × 10^{−8} | 0.233 |

9 | 0.450 | 4.3 × 10^{−6} | 0.129 | 9.7 × 10^{−4} | 1.6 × 10^{−4} | 0.567 | 5.3 × 10^{−5} | 1.6 × 10^{−5} | 0.720 |

10 | 0.433 | 4.3 × 10^{−4} | 0.454 | 6.3 × 10^{−4} | 7.0 × 10^{−6} | 0.193 | 1.4 × 10^{−6} | 1.4 × 10^{−5} | 0.387 |

11 | 0.104 | 1.7 × 10^{−5} | 0.965 | 2.8 × 10^{−5} | 9.0 × 10^{−5} | 0.656 | 1.8 × 10^{−8} | 4.3 × 10^{−8} | 0.701 |

12 | 0.034 | 0.050 | 0.873 | 1.6 × 10^{−4} | 7.2 × 10^{−8} | 0.041 | 8.6 × 10^{−10} | 1.7 × 10^{−7} | 0.159 |

13 | 0.025 | 0.062 | 0.686 | 9.1 × 10^{−7} | 7.9 × 10^{−6} | 0.701 | 9.6 × 10^{−9} | 1.7 × 10^{−5} | 0.009 |

14 | 0.702 | 0.914 | 0.780 | 1.2 × 10^{−7} | 1.5 × 10^{−4} | 0.056 | 4.8 × 10^{−13} | 5.8 × 10^{−9} | 0.020 |

15 | 0.912 | 0.779 | 0.854 | 3.7 × 10^{−5} | 3.0 × 10^{−4} | 0.505 | 3.4 × 10^{−11} | 2.9 × 10^{−7} | 0.020 |

16 | 0.074 | 0.411 | 0.296 | 1.1 × 10^{−4} | 2.7 × 10^{−4} | 0.745 | 2.6 × 10^{−13} | 5.9 × 10^{−10} | 0.070 |

17 | 0.821 | 0.520 | 0.648 | 3.8 × 10^{−5} | 2.3 × 10^{−4} | 0.672 | 5.4 × 10^{−10} | 6.1 × 10^{−8} | 0.309 |

18 | 0.171 | 0.531 | 0.431 | 1.1 × 10^{−5} | 5.9 × 10^{−6} | 0.880 | 2.6 × 10^{−9} | 1.4 × 10^{−8} | 0.729 |

19 | 0.448 | 0.437 | 0.103 | 3.7 × 10^{−5} | 1.4 × 10^{−1} | 0.758 | 2.0 × 10^{−11} | 6.1 × 10^{−11} | 0.947 |

20 | 0.272 | 0.556 | 0.588 | 0.002 | 0.005 | 0.671 | 1.0 × 10^{−9} | 3.9 × 10^{−8} | 0.496 |

**Table 3.**Statistical analysis results of MDE for RR interval of the CHF patients, healthy young, and healthy elder groups. C, E, and Y represent CHF, Elder, and Young, respectively. s denotes scale factor.

MDE Statistical Results for RR Interval Time Series | |||||||||
---|---|---|---|---|---|---|---|---|---|

s | p-Value | ||||||||

$\mathit{N}=100$ | $\mathit{N}=300$ | $\mathit{N}=1000$ | |||||||

C - E | C - Y | E - Y | C - E | C - Y | E - Y | C - E | C - Y | E - Y | |

1 | 2.8 × 10^{−15} | 3.6 × 10^{−26} | 6.9 × 10^{−15} | 2.1 × 10^{−11} | 9.2 × 10^{−26} | 5.5 × 10^{−16} | 1.8 × 10^{−10} | 3.1 × 10^{−26} | 1.1 × 10^{−16} |

2 | 1.7 × 10^{−18} | 7.2 × 10^{−31} | 1.3 × 10^{−13} | 2.7 × 10^{−15} | 7.1 × 10^{−32} | 1.1 × 10^{−14} | 1.7 × 10^{−14} | 1.6 × 10^{−31} | 1.0 × 10^{−16} |

3 | 4.9 × 10^{−21} | 5.5 × 10^{−33} | 2.0 × 10^{−12} | 6.5 × 10^{−18} | 2.3 × 10^{−34} | 2.7 × 10^{−14} | 2.8 × 10^{−17} | 5.6 × 10^{−34} | 3.5 × 10^{−17} |

4 | 1.1 × 10^{−21} | 1.5 × 10^{−33} | 4.3 × 10^{−12} | 1.3 × 10^{−18} | 1.6 × 10^{−34} | 1.3 × 10^{−14} | 3.9 × 10^{−35} | 3.9 × 10^{−35} | 4.7 × 10^{−18} |

5 | 2.5 × 10^{−23} | 3.3 × 10^{−34} | 1.5 × 10^{−11} | 1.2 × 10^{−13} | 9.7 × 10^{−35} | 1.2 × 10^{−14} | 1.8 × 10^{−35} | 1.8 × 10^{−35} | 6.1 × 10^{−17} |

6 | 2.1 × 10^{−23} | 1.7 × 10^{−34} | 4.8 × 10^{−11} | 2.6 × 10^{−20} | 5.7 × 10^{−35} | 2.4 × 10^{−14} | 5.4 × 10^{−35} | 5.4 × 10^{−35} | 2.0 × 10^{−15} |

7 | 3.1 × 10^{−23} | 5.8 × 10^{−34} | 3.0 × 10^{−10} | 4.9 × 10^{−20} | 2.2 × 10^{−34} | 2.4 × 10^{−13} | 1.1 × 10^{−33} | 1.1 × 10^{−33} | 8.1 × 10^{−13} |

8 | 2.1 × 10^{−23} | 8.0 × 10^{−34} | 2.0 × 10^{−9} | 4.6 × 10^{−20} | 3.4 × 10^{−34} | 3.0 × 10^{−12} | 1.6 × 10^{−32} | 1.6 × 10^{−32} | 1.4 × 10^{−10} |

9 | 6.7 × 10^{−23} | 2.1 × 10^{−33} | 4.5 × 10^{−9} | 1.9 × 10^{−20} | 4.4 × 10^{−34} | 9.1 × 10^{−11} | 6.9 × 10^{−32} | 6.9 × 10^{−32} | 1.9 × 10^{−9} |

10 | 6.9 × 10^{−23} | 2.1 × 10^{−33} | 9.6 × 10^{−9} | 3.3 × 10^{−20} | 1.4 × 10^{−33} | 2.9 × 10^{−9} | 9.6 × 10^{−31} | 9.6 × 10^{−31} | 8.9 × 10^{−8} |

11 | 1.7 × 10^{−22} | 3.3 × 10^{−33} | 2.0 × 10^{−8} | 3.8 × 10^{−20} | 2.3 × 10^{−33} | 2.5 × 10^{−8} | 3.3 × 10^{−30} | 3.3 × 10^{−30} | 4.8 × 10^{−6} |

12 | 2.4 × 10^{−22} | 2.9 × 10^{−33} | 2.5 × 10^{−8} | 4.6 × 10^{−20} | 1.5 × 10^{−32} | 1.6 × 10^{−7} | 5.9 × 10^{−29} | 5.9 × 10^{−29} | 1.8 × 10^{−5} |

13 | 2.6 × 10^{−22} | 1.1 × 10^{−32} | 5.7 × 10^{−8} | 1.9 × 10^{−20} | 4.6 × 10^{−32} | 7.4 × 10^{−7} | 2.4 × 10^{−27} | 2.4 × 10^{−27} | 1.4 × 10^{−4} |

14 | 8.0 × 10^{−22} | 7.9 × 10^{−33} | 1.0 × 10^{−7} | 8.8 × 10^{−20} | 4.4 × 10^{−31} | 2.1 × 10^{−6} | 2.8 × 10^{−25} | 2.8 × 10^{−25} | 6.0 × 10^{−4} |

15 | 3.2 × 10^{−21} | 4.5 × 10^{−32} | 2.3 × 10^{−7} | 3.5 × 10^{−19} | 3.8 × 10^{−30} | 4.8 × 10^{−6} | 4.0 × 10^{−23} | 4.0 × 10^{−23} | 0.001 |

16 | 1.0 × 10^{−20} | 1.1 × 10^{−31} | 3.8 × 10^{−7} | 4.7 × 10^{−18} | 1.1 × 10^{−28} | 4.8 × 10^{−6} | 7.6 × 10^{−21} | 7.6 × 10^{−21} | 8.8 × 10^{−4} |

17 | 3.5 × 10^{−20} | 2.4 × 10^{−31} | 4.4 × 10^{−7} | 1.6 × 10^{−17} | 2.7 × 10^{−27} | 1.7 × 10^{−5} | 2.2 × 10^{−18} | 2.2 × 10^{−18} | 0.004 |

18 | 7.9 × 10^{−20} | 3.4 × 10^{−31} | 9.6 × 10^{−7} | 1.6 × 10^{−17} | 4.0 × 10^{−27} | 3.7 × 10^{−5} | 1.1 × 10^{−16} | 1.1 × 10^{−16} | 0.005 |

19 | 2.1 × 10^{−19} | 1.4 × 10^{−30} | 1.4 × 10^{−6} | 7.6 × 10^{−17} | 1.8 × 10^{−26} | 1.2 × 10^{−9} | 5.8 × 10^{−16} | 5.8 × 10^{−16} | 0.006 |

20 | 5.3 × 10^{−19} | 1.9 × 10^{−30} | 4.0 × 10^{−6} | 1.7 × 10^{−16} | 7.0 × 10^{−26} | 1.1 × 10^{−5} | 2.4 × 10^{−14} | 2.4 × 10^{−14} | 0.015 |

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**MDPI and ACS Style**

Lee, D.-Y.; Choi, Y.-S. Multiscale Distribution Entropy Analysis of Short-Term Heart Rate Variability. *Entropy* **2018**, *20*, 952.
https://doi.org/10.3390/e20120952

**AMA Style**

Lee D-Y, Choi Y-S. Multiscale Distribution Entropy Analysis of Short-Term Heart Rate Variability. *Entropy*. 2018; 20(12):952.
https://doi.org/10.3390/e20120952

**Chicago/Turabian Style**

Lee, Dae-Young, and Young-Seok Choi. 2018. "Multiscale Distribution Entropy Analysis of Short-Term Heart Rate Variability" *Entropy* 20, no. 12: 952.
https://doi.org/10.3390/e20120952