# Multivariable Fuzzy Measure Entropy Analysis for Heart Rate Variability and Heart Sound Amplitude Variability

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## Abstract

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## 1. Introduction

## 2. Multivariate Entropy Measures

#### 2.1. Multivariate Sample Entropy (mvSE)

- (1)
- For a p-variate time series ${\left\{{x}_{k,i}\right\}}_{i=1}^{N}$, $k=1,2,\cdots ,p$, where $N$ is the number of samples in each variate, firstly normalize each time series for $k=1,2,\cdots ,p$, and then form the composite delay vector using a composite delay factor based on the multivariate embedded reconstruction:$${X}_{i}^{m}=\left[{x}_{1,i},{x}_{1,i+{\tau}_{1}},\cdots ,{x}_{1,i+\left({m}_{1}-1\right){\tau}_{1}},{x}_{2,i},{x}_{2,i+{\tau}_{2}},\cdots ,{x}_{2,i+\left({m}_{2}-1\right){\tau}_{2}},\cdots ,{x}_{p,i},{x}_{p,i+{\tau}_{p}},\cdots ,{x}_{p,i+\left({m}_{p}-1\right){\tau}_{p}}\right],$$
- (2)
- Define the distance ${d}_{i,j}^{m}$ between any two composite delay vectors ${X}_{i}^{m}$ and ${X}_{j}^{m}$ as the maximum norm, that is,$${d}_{i,j}^{m}=d[{X}_{i}^{m},{X}_{j}^{m}]=\underset{l=1,2,\cdots ,m}{\mathrm{max}}\left(\left|x\left(i+l-1\right)-x\left(j+l-1\right)\right|\right).$$
- (3)
- For a given vector ${X}_{i}^{m}$ and a threshold $r$, count the number of instances ${P}_{i}$ where ${d}_{i,j}^{m}\le r$, $j\ne i$, and then calculate the frequency of occurrence, ${B}_{i}^{m}\left(r\right)={\left(N-n-1\right)}^{-1}{P}_{i}$, and define a global quantity ${B}^{m}\left(r\right)={\left(N-n\right)}^{-1}{\displaystyle \sum}_{i=1}^{N-n}{B}_{i}^{m}\left(r\right)$.
- (4)
- Extend the dimensionality of the multivariate delay vector in Equation (1) from $m$ to $m+1$. This can be performed in $p$ different ways, as the system can evolve to any space with $M=\left[{m}_{1},{m}_{2},\cdots ,{m}_{k}+1,\cdots ,{m}_{p}\right]$ ($k=1,2,\cdots ,p$). Thus, a total of $p$ vectors ${X}_{i}^{m+1}\in {R}^{m+1}$ are obtained. The k-th vector ${X}_{i}^{m+1}$ is:$${X}_{i}^{m+1}=[{x}_{1,i},{x}_{1,i+{\tau}_{1}},\cdots ,{x}_{1,i+\left({m}_{1}-1\right){\tau}_{1}},{x}_{2,i},{x}_{2,i+{\tau}_{2}},\cdots ,{x}_{2,i+\left({m}_{2}-1\right){\tau}_{2}},\cdots ,{x}_{k,i},{x}_{k,i+{\tau}_{k}},\phantom{\rule{0ex}{0ex}}\cdots ,{x}_{k,i+\left({m}_{k}-1\right){\tau}_{k}},{x}_{k,i+{m}_{k}{\tau}_{k}},\cdots ,{x}_{p,i},{x}_{p,i+{\tau}_{p}},\cdots ,{x}_{p,i+\left({m}_{p}-1\right){\tau}_{p}}].$$
- (5)
- For a given ${X}_{i}^{m+1}$, count the number of instances ${Q}_{i}$, where ${d}_{i,j}^{m+1}\le r$, $j\ne i$, and then calculate the frequency of occurrence, ${B}_{i}^{m+1}\left(r\right)={\left(p\left(N-n\right)-1\right)}^{-1}{Q}_{i}$, and define ${B}^{m+1}\left(r\right)={\left(p\left(N-n\right)\right)}^{-1}{\displaystyle \sum}_{i=1}^{p\left(N-n\right)}{B}_{i}^{m+1}\left(r\right)$.
- (6)
- Finally, mvSE is defined by$$\mathrm{mvSE}\left(M,\tau ,r,N\right)=-\mathrm{ln}\left[\frac{{B}^{m+1}\left(r\right)}{{B}^{m}\left(r\right)}\right].$$

#### 2.2. Multivariate Fuzzy Measure Entropy (mvFME)

- (1)
- For a $p$-variate time series ${\left\{{x}_{k,i}\right\}}_{i=1}^{N}$, $k=1,2,\cdots ,p$, where $N$ is the number of samples in each variate, firstly normalize each time series for $k=1,2,\cdots ,p$, and then form the local composite delay vector and global composite delay vector using a composite delay factor based on the multivariate embedded reconstruction:$${\overline{X}}_{i}^{m}=\left[{\overline{x}}_{1,i},{\overline{x}}_{1,i+{\tau}_{1}},\cdots ,{\overline{x}}_{1,i+\left({m}_{1}-1\right){\tau}_{1}},{\overline{x}}_{2,i},{\overline{x}}_{2,i+{\tau}_{2}},\cdots ,{\overline{x}}_{2,i+\left({m}_{2}-1\right){\tau}_{2}},\cdots ,{\overline{x}}_{p,i},{\overline{x}}_{p,i+{\tau}_{p}},\cdots ,{\overline{x}}_{p,i+\left({m}_{p}-1\right){\tau}_{p}}\right],$$$${\stackrel{=}{X}}_{i}^{m}=\left[{\stackrel{=}{x}}_{1,i},{\stackrel{=}{x}}_{1,i+{\tau}_{1}},\cdots ,{\stackrel{=}{x}}_{1,i+\left({m}_{1}-1\right){\tau}_{1}},{\stackrel{=}{x}}_{2,i},{\stackrel{=}{x}}_{2,i+{\tau}_{2}},\cdots ,{\stackrel{=}{x}}_{2,i+\left({m}_{2}-1\right){\tau}_{2}},\cdots ,{\stackrel{=}{x}}_{p,i},{\stackrel{=}{x}}_{p,i+{\tau}_{p}},\cdots ,{\stackrel{=}{x}}_{p,i+\left({m}_{p}-1\right){\tau}_{p}}\right],$$
- (2)
- Define the distance $d{L}_{i,j}^{m}$ between any two local composite delay vectors ${\overline{X}}_{i}^{m}$ and ${\overline{X}}_{j}^{m}$, and the distance $d{G}_{i,j}^{m}$ between any two global composite delay vectors ${\stackrel{=}{X}}_{i}^{m}$ and ${\stackrel{=}{X}}_{j}^{m}$, as the maximum norm, that is,$$d{L}_{i,j}^{m}=d\left[{\overline{X}}_{i}^{m},{\overline{X}}_{j}^{m}\right]=\underset{l=1,2,\cdots ,m}{\mathrm{max}}\left(\left|\overline{x}\left(i+l-1\right)-\overline{x}\left(j+l-1\right)\right|\right),$$$$d{G}_{i,j}^{m}=d\left[{\stackrel{=}{X}}_{i}^{m},{\stackrel{=}{X}}_{j}^{m}\right]=\underset{l=1,2,\cdots ,m}{\mathrm{max}}\left(\left|\stackrel{=}{x}\left(i+l-1\right)-\stackrel{=}{x}\left(j+l-1\right)\right|\right),$$
- (3)
- Given the parameters’ local vector similarity weight ${n}_{L}$, local tolerance threshold ${r}_{L}$, global vector similarity weight ${n}_{G}$ and global tolerance threshold ${r}_{G}$, calculate the similarity degree $D{L}_{i,j}^{m}\left({n}_{L},{r}_{L}\right)$ between the local composite delay vectors ${\overline{X}}_{i}^{m}$ and ${\overline{X}}_{j}^{m}$ by the fuzzy function $\mu L(d{L}_{i,j}^{m},{n}_{L},{r}_{L})$, as well as calculate the similarity degree $D{G}_{i,j}^{m}\left({n}_{L},{r}_{L}\right)$ between the global composite delay vectors ${\stackrel{=}{X}}_{i}^{m}$ and ${\stackrel{=}{X}}_{j}^{m}$ by the fuzzy function $\mu G(d{G}_{i,j}^{m},{n}_{G},{r}_{G})$:$$D{L}_{i,j}^{m}\left({n}_{L},{r}_{L}\right)=\mu L\left(d{L}_{i,j}^{m},{n}_{L},{r}_{L}\right)=\mathrm{exp}\left(-\frac{{\left(d{L}_{i,j}^{m}\right)}^{{n}_{L}}}{{r}_{L}}\right),$$$$D{G}_{i,j}^{m}\left({n}_{G},{r}_{G}\right)=\mu G\left(d{G}_{i,j}^{m},{n}_{G},{r}_{G}\right)=\mathrm{exp}\left(-\frac{{\left(d{G}_{i,j}^{m}\right)}^{{n}_{G}}}{{r}_{G}}\right).$$$$\varnothing {L}^{m}\left({n}_{L},{r}_{L}\right)=\frac{1}{N-n}{{\displaystyle \sum}}_{i=1}^{N-n}(\frac{1}{N-n-1}{{\displaystyle \sum}}_{j=1,j\ne i}^{N-n}D{L}_{i,j}^{m}\left({n}_{L},{r}_{L}\right)),$$$$\varnothing {G}^{m}\left({n}_{G},{r}_{G}\right)=\frac{1}{N-n}{{\displaystyle \sum}}_{i=1}^{N-n}(\frac{1}{N-n-1}{{\displaystyle \sum}}_{j=1,j\ne i}^{N-n}D{G}_{i,j}^{m}\left({n}_{G},{r}_{G}\right)).$$
- (4)
- Extend the dimensionality of the multivariate delay vectors in Equations (5) and (6) from $m$ to $m+1$. This can be performed in $p$ different ways, as the system can evolve to any space with $M=\left[{m}_{1},{m}_{2},\cdots ,{m}_{k}+1,\cdots ,{m}_{p}\right]$ ($k=1,2,\cdots ,p$). Thus, a total of $p$ vectors ${\overline{X}}_{i}^{m+1}\in {R}^{m+1}$ and a total of $p$ vectors ${\stackrel{=}{X}}_{i}^{m+1}\in {R}^{m+1}$ are obtained. The kth vectors ${\overline{X}}_{i}^{m+1}$ and ${\stackrel{=}{X}}_{i}^{m+1}$ are respectively:$${\overline{X}}_{i}^{m+1}=[{\overline{x}}_{1,i},{\overline{x}}_{1,i+{\tau}_{1}},\cdots ,{\overline{x}}_{1,i+\left({m}_{1}-1\right){\tau}_{1}},{\overline{x}}_{2,i},{\overline{x}}_{2,i+{\tau}_{2}},\cdots ,{\overline{x}}_{2,i+\left({m}_{2}-1\right){\tau}_{2}},\cdots ,{\overline{x}}_{k,i},{\overline{x}}_{k,i+{\tau}_{k}},\phantom{\rule{0ex}{0ex}}\cdots ,{\overline{x}}_{k,i+\left({m}_{k}-1\right){\tau}_{k}},{\overline{x}}_{k,i+{m}_{k}{\tau}_{k}},\cdots ,{\overline{x}}_{p,i},{\overline{x}}_{p,i+{\tau}_{p}},\cdots ,{\overline{x}}_{p,i+\left({m}_{p}-1\right){\tau}_{p}}]$$$${\stackrel{=}{X}}_{i}^{m+1}=[{\stackrel{=}{x}}_{1,i},{\stackrel{=}{x}}_{1,i+{\tau}_{1}},\cdots ,{\stackrel{=}{x}}_{1,i+\left({m}_{1}-1\right){\tau}_{1}},{\stackrel{=}{x}}_{2,i},{\stackrel{=}{x}}_{2,i+{\tau}_{2}},\cdots ,{\stackrel{=}{x}}_{2,i+\left({m}_{2}-1\right){\tau}_{2}},\cdots ,{\stackrel{=}{x}}_{k,i},{\stackrel{=}{x}}_{k,i+{\tau}_{k}},\phantom{\rule{0ex}{0ex}}\cdots ,{\stackrel{=}{x}}_{k,i+\left({m}_{k}-1\right){\tau}_{k}},{\stackrel{=}{x}}_{k,i+{m}_{k}{\tau}_{k}},\cdots ,{\stackrel{=}{x}}_{p,i},{\stackrel{=}{x}}_{p,i+{\tau}_{p}},\cdots ,{\stackrel{=}{x}}_{p,i+\left({m}_{p}-1\right){\tau}_{p}}]$$
- (5)
- Similarly, define the function $\varnothing {L}^{m+1}\left({n}_{L},{r}_{L}\right)$ for the local composite delay vectors ${\overline{X}}_{i}^{m+1}$ and ${\overline{X}}_{j}^{m+1}$ and the function $\varnothing {G}^{m+1}\left({n}_{G},{r}_{G}\right)$ for the global composite delay vectors ${\stackrel{=}{X}}_{i}^{m+1}$ and ${\stackrel{=}{X}}_{j}^{m+1}$ as:$$\varnothing {L}^{m+1}\left({n}_{L},{r}_{L}\right)=\frac{1}{p\left(N-n\right)}{{\displaystyle \sum}}_{i=1}^{p\left(N-n\right)}(\frac{1}{p\left(N-n\right)-1}{{\displaystyle \sum}}_{j=1,j\ne i}^{p\left(N-n\right)}D{L}_{i,j}^{m+1}\left({n}_{L},{r}_{L}\right)),$$$$\varnothing {G}^{m+1}\left({n}_{G},{r}_{G}\right)=\frac{1}{p\left(N-n\right)}{{\displaystyle \sum}}_{i=1}^{p\left(N-n\right)}(\frac{1}{p\left(N-n\right)-1}{{\displaystyle \sum}}_{j=1,j\ne i}^{p\left(N-n\right)}D{G}_{i,j}^{m+1}\left({n}_{G},{r}_{G}\right)).$$
- (6)
- Then, the local multivariate fuzzy entropy (mvFEL) and global multivariate fuzzy entropy (mvFEG) are defined by:$$\mathrm{mvFEL}\left(M,\tau ,{n}_{L},{r}_{L},N\right)=-\mathrm{ln}\left[\varnothing {L}^{m+1}\left({n}_{L},{r}_{L}\right)/\varnothing {L}^{m}\left({n}_{L},{r}_{L}\right)\right],$$$$\mathrm{mvFEG}\left(M,\tau ,{n}_{G},{r}_{G},N\right)=-\mathrm{ln}\left[\frac{\varnothing {G}^{m+1}\left({n}_{G},{r}_{G}\right)}{\varnothing {G}^{m}\left({n}_{G},{r}_{G}\right)}\right].$$
- (7)
- Finally, mvFME is defined by$$\mathrm{mvFME}\left(M,\tau ,{n}_{L},{r}_{L},{n}_{G},{r}_{G},N\right)=\mathrm{mvFEL}\left(M,\tau ,{n}_{L},{r}_{L},N\right)+\mathrm{mvFEG}\left(M,\tau ,{n}_{G},{r}_{G},N\right).$$
_{L}= 2 and global vector similarity weight n_{G}= 2, and the local tolerance threshold r_{L}was set equal to the global threshold r_{G}, i.e., r_{L}= r_{G}= r. Thus, Equation (19) becomes:$$\mathrm{mvFME}\left(M,\tau ,r,N\right)=\mathrm{mvFEL}\left(M,\tau ,r,N\right)+\mathrm{mvFEG}\left(M,\tau ,r,N\right).$$

## 3. Experiment Design

#### 3.1. Simulation Signals

#### 3.2. Cardiovascular Signals

#### 3.3. Statistical Analysis

## 4. Results

#### 4.1. Results on Simulation Signals

#### 4.2. Results on Cardiovascular Signals

## 5. Discussions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Simultaneously recorded electrocardiogram (ECG) and three phonocardiogram (PCG) signals (from top to bottom are ECG and PCGs recorded from aortic, mitral and tricuspid areas, respectively). The detected R-wave peaks are denoted as “●”, the S1 and S2 heart sounds were identified using Springer’s hidden semi-Markov model (HSMM) method.

**Figure 2.**Examples of R wave peak (RR) interval, S1 and S2 amplitude series from the ECG and aortic PCG signals. Sub-figures (

**A1**–

**A3**) show the three cardiovascular time series from the rest state; and sub-figures (

**B1**–

**B3**) show the corresponding time series after stair climbing. For each time series, only the first 200 beats were shown.

**Figure 3.**Dependence of the multivariate entropy measures (multivariate Sample Entropy (mvSE), local multivariate fuzzy entropy (mvFEL), global multivariate fuzzy entropy (mvFEG) and multivariate fuzzy measure entropy (mvFME)) on the coupling degree c when applied to the coupled Gaussian noise signals. (

**A**) univariate analysis using time series $x$ only; (

**B**) bivariate analysis using time series $x$ and $y$; and (

**C**) trivariate analysis using time series $x$, $y$ and $z$.

**Figure 4.**Statistical results of mvSE, mvFEL, mvFEG and mvFME between the rest and after stair climbing states by analyzing the multivariate cardiovascular time series, i.e., univariate analysis for RR interval, S1 and S2 amplitude series, respectively, bivariate analysis for each two of the three time series and trivariate analysis for the three time series. (

**A**) PCG signal from aortic area; (

**B**) PCG signal from mitral area; and (

**C**) PCG signal from the tricuspid area. Rest: rest state; Climb: after stair climbing state; RR: RR interval time series; S1: S1 heart sound amplitude series; S2: S2 heart sound amplitude series; *: statistical significance p < 0.05; **: statistical significance p < 0.01.

Variable | Value |
---|---|

Age (year) | 24.2 ± 1.9 |

Height (cm) | 174 ± 4 |

Weight (kg) | 64 ± 7 |

Heart rate (beats/min) | 69 ± 9 |

Systolic blood pressure (mmHg) | 121 ± 9 |

Diastolic blood pressure (mmHg) | 65 ± 7 |

**Table 2.**Comparison of the computation time for the four multivariate entropy measures (multivariate Sample Entropy (mvSE), local multivariate fuzzy entropy (mvFEL), global multivariate fuzzy entropy (mvFEG) and multivariate fuzzy measure entropy (mvFME)) on simulation Gaussian time series.

Gaussian Time Series | Time (s) | |||
---|---|---|---|---|

mvSE | mvFEL | mvFEG | mvFME | |

Univariate analysis | 6.58 | 16.67 | 16.61 | 33.28 |

Bivariate analysis | 19.42 | 51.20 | 51.03 | 102.23 |

Trivariate analysis | 44.01 | 108.84 | 108.47 | 217.31 |

**Table 3.**Statistical results of mvSE and mvFME between the rest and after stair climbing states by analyzing the univariate, bivariate and trivariate cardiovascular time series from electrocardiogram (ECG) and three phonocardiogram (PCG) signals, respectively.

Signals | Time Series | mvSE | mvFME | ||||
---|---|---|---|---|---|---|---|

Rest | Climb | p-Value | Rest | Climb | p-Value | ||

ECG + aortic PCG | RR | 2.15 ± 0.38 | 1.11 ± 0.30 | 6 × 10^{−9} | 2.17 ± 0.34 | 0.93 ± 0.37 | 3 × 10^{−10} |

S1 | 2.26 ± 0.26 | 1.53 ± 0.37 | 6 × 10^{−7} | 2.60 ± 0.18 | 1.65 ± 0.50 | 3 × 10^{−7} | |

S2 | 2.33 ± 0.32 | 2.16 ± 0.29 | 0.1 | 2.59 ± 0.23 | 2.32 ± 0.31 | 3 × 10^{−3} | |

RR & S1 | 1.52 ± 0.20 | 1.13 ± 0.13 | 2 × 10^{−6} | 1.81 ± 0.13 | 1.16 ± 0.19 | 9 × 10^{−11} | |

RR & S2 | 1.46 ± 0.19 | 1.22 ± 0.19 | 9 × 10^{−4} | 1.78 ± 0.19 | 1.32 ± 0.11 | 1 × 10^{−8} | |

S1 & S2 | 1.60 ± 0.25 | 1.16 ± 0.22 | 6 × 10^{−6} | 2.00 ± 0.21 | 1.38 ± 0.30 | 2 × 10^{−9} | |

RR & S1 & S2 | 1.04 ± 0.42 | 1.06 ± 0.09 | 0.8 | 1.50 ± 0.21 | 1.09 ± 0.16 | 1 × 10^{−8} | |

ECG + mitral PCG | RR | 2.15 ± 0.38 | 1.11 ± 0.30 | 6 × 10^{−9} | 2.17 ± 0.34 | 0.93 ± 0.37 | 3 × 10^{−10} |

S1 | 2.16 ± 0.42 | 1.66 ± 0.62 | 5 × 10^{−3} | 2.43 ± 0.25 | 1.81 ± 0.59 | 2 × 10^{−4} | |

S2 | 2.27 ± 0.38 | 2.01 ± 0.57 | 0.1 | 2.50 ± 0.34 | 2.27 ± 0.48 | 0.1 | |

RR & S1 | 1.52 ± 0.14 | 1.14 ± 0.16 | 2 × 10^{−8} | 1.80 ± 0.20 | 1.18 ± 0.20 | 3 × 10^{−11} | |

RR & S2 | 1.44 ± 0.25 | 1.18 ± 0.15 | 5 × 10^{−4} | 1.76 ± 0.23 | 1.32 ± 0.14 | 3 × 10^{−6} | |

S1 & S2 | 1.52 ± 0.26 | 1.20 ± 0.43 | 9 × 10^{−3} | 1.96 ± 0.27 | 1.46 ± 0.44 | 3 × 10^{−4} | |

RR & S1 & S2 | 0.98 ± 0.27 | 1.06 ± 0.17 | 0.3 | 1.50 ± 0.25 | 1.12 ± 0.19 | 2 × 10^{−5} | |

ECG + tricuspid PCG | RR | 2.15 ± 0.38 | 1.11 ± 0.30 | 6 × 10^{−9} | 2.17 ± 0.34 | 0.93 ± 0.37 | 3 × 10^{−10} |

S1 | 2.19 ± 0.37 | 1.75 ± 0.48 | 7 × 10^{−3} | 2.47 ± 0.37 | 1.92 ± 0.49 | 2 × 10^{−3} | |

S2 | 2.25 ± 0.29 | 2.05 ± 0.42 | 0.1 | 2.47 ± 0.32 | 2.37 ± 0.39 | 0.3 | |

RR & S1 | 1.47 ± 0.23 | 1.17 ± 0.16 | 2 × 10^{−6} | 1.81 ± 0.22 | 1.22 ± 0.16 | 2 × 10^{−9} | |

RR & S2 | 1.47 ± 0.18 | 1.16 ± 0.13 | 1 × 10^{−6} | 1.72 ± 0.21 | 1.33 ± 0.12 | 3 × 10^{−8} | |

S1 & S2 | 1.54 ± 0.23 | 1.27 ± 0.34 | 9.9 × 10^{−3} | 1.93 ± 0.31 | 1.58 ± 0.33 | 2 × 10^{−3} | |

RR & S1 & S2 | 1.09 ± 0.34 | 1.10 ± 0.14 | 0.97 | 1.47 ± 0.24 | 1.18 ± 0.15 | 1 × 10^{−4} |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhao, L.; Wei, S.; Tang, H.; Liu, C. Multivariable Fuzzy Measure Entropy Analysis for Heart Rate Variability and Heart Sound Amplitude Variability. *Entropy* **2016**, *18*, 430.
https://doi.org/10.3390/e18120430

**AMA Style**

Zhao L, Wei S, Tang H, Liu C. Multivariable Fuzzy Measure Entropy Analysis for Heart Rate Variability and Heart Sound Amplitude Variability. *Entropy*. 2016; 18(12):430.
https://doi.org/10.3390/e18120430

**Chicago/Turabian Style**

Zhao, Lina, Shoushui Wei, Hong Tang, and Chengyu Liu. 2016. "Multivariable Fuzzy Measure Entropy Analysis for Heart Rate Variability and Heart Sound Amplitude Variability" *Entropy* 18, no. 12: 430.
https://doi.org/10.3390/e18120430