# Multi-Chaotic Analysis of Inter-Beat (R-R) Intervals in Cardiac Signals for Discrimination between Normal and Pathological Classes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Clinical Datasets

#### 2.2. Statistical Methods

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**The dependencies of E1(m) versus m for R-R interval time series data for p-norm = 0.1, 0.5, and 1–5.

## References

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**Figure 1.**This plot shows the relationship between $\mathrm{ln}({F}_{q}(\Delta t))$ and $\mathrm{ln}(\Delta t)$ for random time series data for q = −1, 2, and 5.

**Figure 2.**The dependencies of<ln(divergence)> versus time for the logistic map for p-norm = 0.1, 2, and 6.

**Figure 3.**(

**a**) Generalized largest Lyapunov exponents GLLE(p) vs. order p for healthy (hs), congestive heart failure (chf), and atrial fibrillation (af) groups. (

**b**) The same values for p = 3, 4, and 5. The error bars represent the standard error of the mean.

**Table 1.**The chaotic dynamical systems that were used to evaluate the generalized largest Lyapunov exponents (GLLE).

System | Equations | Parameters | $\mathbf{\Delta}\mathit{t}(\mathit{s})$ | Theoretical $\mathit{L}\mathit{L}\mathit{E}$ [20] |
---|---|---|---|---|

Logistic | ${x}_{i+1}=\mu {x}_{i}(1-{x}_{i})$ | $\mu $ = 4.0 | 1 | 0.69 |

Henon | ${x}_{i+1}=1-a{x}_{i}^{2}+{y}_{i}$ ${y}_{i+1}=b{x}_{i}$ | a = 1.4 b = 0.3 | 1 | 0.42 |

Lorenz | ${x}^{\prime}=\sigma (y-x)$ ${y}^{\prime}=x(R-z)-y$ ${z}^{\prime}=xy-bz$ | $\sigma $ = 10.0 R = 28 b = 8/3 | 0.01 | 1.50 |

Rossler | ${x}^{\prime}=-y-z$ ${y}^{\prime}=x+ay$ ${z}^{\prime}=b+z(x-c)$ | a = 0.15 b = 0.20 c = 10.0 | 0.1 | 0.09 |

p-Norm | Logistic, GLLE(p) | Henon, GLLE(p) | Lorenz, GLLE(p) | Rossler, GLLE(p) |
---|---|---|---|---|

0.1 | 0.69 | 0.42 | 1.44 | 0.07 |

0.5 | 0.69 | 0.43 | 1.50 | 0.09 |

1 | 0.69 | 0.43 | 1.51 | 0.09 |

2 | 0.69 | 0.42 | 1.52 | 0.09 |

3 | 0.67 | 0.42 | 1.51 | 0.09 |

4 | 0.62 | 0.39 | 1.53 | 0.09 |

5 | 0.49 | 0.34 | 1.52 | 0.09 |

6 | 0.38 | 0.30 | 1.30 | 0.06 |

7 | 0.29 | 0.25 | 1.25 | 0.06 |

8 | 0.25 | 0.22 | 1.32 | 0.06 |

9 | 0.22 | 0.18 | 1.26 | 0.06 |

10 | 0.22 | 0.16 | 1.27 | 0.06 |

$\Delta $W | 0.47 | 0.27 | 0.28 | 0.02 |

**Table 3.**Calculation results of the estimation of the spectrum width ΔW for chaotic dynamical systems with the addition of a random component, having a normal distribution with a zero mean and different standard deviations, to the x-coordinate time series.

Standard Deviation | Logistic, ΔW | Henon, ΔW | Lorenz, ΔW | Rossler, ΔW |
---|---|---|---|---|

0.001 | 0.50 | 0.31 | 0.26 | 0.01 |

0.01 | 0.28 | 0.23 | 0.21 | 0.01 |

0.05 | 0.20 | 0.17 | 0.21 | 0.01 |

**Table 4.**Median of the largest Lyapunov exponent GLLE(2) and the estimation of the spectrum width $\Delta $W (Me—median, ${Q}_{1}$—first quartile, and ${Q}_{3}$—third quartile).

Group | Number | GLLE(2), $\mathbf{Me}({\mathit{Q}}_{1}$$\u2013{\mathit{Q}}_{3})$ | $\mathbf{\Delta}$W $\mathbf{Me}({\mathit{Q}}_{1}$$\u2013{\mathit{Q}}_{3})$ |
---|---|---|---|

hs | 54 | 0.11(0.10–0.13) ${}^{a}$ | 1.28(0.78–1.95) ${}^{c}$ |

chf | 44 | 0.15(0.12–0.20) ${}^{a}$^{,}${}^{b}$ | 2.61(1.70–4.23) ${}^{c}$^{,}${}^{d}$ |

af | 25 | 0.08(0.07–0.10) ${}^{a}$^{,}${}^{b}$ | 0.42(0.26–0.79) ${}^{c}$^{,}${}^{d}$ |

^{a}Significant difference between the GLLE(2) of healthy (hs) and pathological groups, p-value < 0.01 using the Kruskal–Wallis test;

^{b}significant difference between the GLLE(2) of congestive heart failure (chf) and atrial fibrillation (af) groups, p-value < 0.01, where a Kruskal–Wallis test was conducted;

^{c}significant difference between the $\Delta $W of healthy (hs) and pathological groups, p-value < 0.01, using the Kruskal–Wallis test;

^{d}significant difference between the $\Delta $W of congestive heart failure (chf) and atrial fibrillation (af) groups, p-value < 0.01, using the Kruskal–Wallis test.

**Table 5.**Median of the correlation dimension ${D}_{2}$ (Me—median, ${Q}_{1}$—first quartile, and ${Q}_{3}$—third quartile).

Group | Number | ${\mathit{D}}_{2},$ $\mathbf{Me}({\mathit{Q}}_{1}$$\u2013{\mathit{Q}}_{3})$ |
---|---|---|

hs | 54 | 0.57(0.50–0.66) ${}^{a}$ |

chf | 44 | 0.55(0.17–0.85) ${}^{b}$ |

af | 25 | 0.93(0.69–1.00) ${}^{a}$^{,}${}^{b}$ |

^{a}Significant difference betweenthe ${D}_{2}$ of healthy (hs) and atrial fibrillation (af) groups, p-value < 0.01, using the Kruskal–Wallis test, and

^{b}significant difference betweenthe ${D}_{2}$ of congestive heart failure (chf) and atrial fibrillation (af) groups, p-value < 0.01, using the Kruskal–Wallis test.

Value | Regression Coefficients b ± m | p-Value | Odds Ratio (95% CI) |
---|---|---|---|

GLLE_0.1 | 1.02 ± 0.28 | <0.01 | 2.78 (1.59–4.84) |

GLLE_0.5 | −5.18 ± 2.11 | <0.01 | 5.62 × 10${}^{-3}$(8.88 × 10${}^{-5}$ − 3.55 × 10${}^{-1}$) |

GLLE_1 | −7.09 ± 3.53 | <0.01 | 8.29 × 10${}^{-4}$(8.11 × 10${}^{-7}$ − 8.47 × 10${}^{-1}$) |

GLLE_4 | 105.39 ± 29.21 | <0.01 | 5.92 × 10${}^{45}$(7.96 × 10${}^{20}$ − 4.41 × 10${}^{70}$) |

GLLE_5 | −95.23 ± 26.57 | <0.01 | 4.37 × 10${}^{-42}$(1.0 × 10${}^{-64}$ − 1.84 × 10${}^{-19}$) |

Constant | −0.45 ± 1.89 | <0.01 |

Classification Results | Set | |||
---|---|---|---|---|

Training | Testing | |||

Classification | ||||

Healthy Group | Pathological Group | Healthy Group | Pathological Group | |

Correct | 31 | 39 | 11 | 20 |

Incorrect | 8 | 7 | 4 | 3 |

Total cases | 39 | 36 | 15 | 23 |

Value | Regression Coefficients b ± m | p-Value | Odds Ratio (95% CI) |
---|---|---|---|

GLLE_1 | −4.851 ± 1.95 | <0.01 | 7.82 × 10${}^{-3}$(1.73 × 10${}^{-4}$ − 3.55 × 10${}^{-1}$) |

GLLE_4 | 75.71 ± 31.29 | <0.01 | 7.62 × 10${}^{32}$(1.77 × 10${}^{6}$ − 3.28 × 10${}^{59}$) |

GLLE_5 | −70.84 ± 29.37 | <0.01 | 1.71 × 10${}^{-31}$(1.72 × 10${}^{-56}$ − 1.69 × 10${}^{-6}$) |

Constant | 3.34 ± 3.01 | <0.01 |

Classification Results | Set | |
---|---|---|

Congestive Heart Failure Group | Atrial Fibrillation Group | |

Correct | 40 | 19 |

Incorrect | 4 | 6 |

Total cases | 44 | 25 |

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**MDPI and ACS Style**

Gorshkov, O.; Ombao, H. Multi-Chaotic Analysis of Inter-Beat (R-R) Intervals in Cardiac Signals for Discrimination between Normal and Pathological Classes. *Entropy* **2021**, *23*, 112.
https://doi.org/10.3390/e23010112

**AMA Style**

Gorshkov O, Ombao H. Multi-Chaotic Analysis of Inter-Beat (R-R) Intervals in Cardiac Signals for Discrimination between Normal and Pathological Classes. *Entropy*. 2021; 23(1):112.
https://doi.org/10.3390/e23010112

**Chicago/Turabian Style**

Gorshkov, Oleg, and Hernando Ombao. 2021. "Multi-Chaotic Analysis of Inter-Beat (R-R) Intervals in Cardiac Signals for Discrimination between Normal and Pathological Classes" *Entropy* 23, no. 1: 112.
https://doi.org/10.3390/e23010112