# Multivariate and Multiscale Complexity of Long-Range Correlated Cardiovascular and Respiratory Variability Series

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Measures of Complexity in Linear Multivariate Stochastic processes

#### 2.2. Linear Multivariate Stochastic Processes with Long Range Correlations

#### 2.3. Multiscale Complexity of VARFI Processes

## 3. Application to Cardiovascular Variability Processes

#### 3.1. Experimental Protocol

#### 3.2. Data Analysis

#### 3.3. Statistical Analysis

`lme4`[37] and

`emmeans`[38] of the

`R`software [39] were used to build the model and to compute EMM, respectively.

## 4. Results

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

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**Figure 1.**Distribution across subjects of the multivariate complexity measure ${\overline{C}}_{\mathbf{X}}$ as a function of the time scale $\tau $ for eVAR (first row) and eVARFI (second row), for SU${}_{1}$ vs. UP (left column) and SU${}_{2}$ vs. MA (right column).

**Figure 2.**Distribution across subjects of the univariate (

**a**), bivariate (

**b**) with SAP and (

**c**) with RESP, and multivariate (

**d**) complexity measures as a function of the time scale τ for eVAR (first row) and eVARFI (second row), for SU1 vs. UP (left column) and SU2 vs. MA (right column).

**Figure 3.**Distribution of the multivariate complexity measure ${\overline{C}}_{\mathbf{X}}$, depicted as boxplot (mean and confidence intervals, yellow filled box; standard deviation, black vertical line) and original values (dots) for selected frequencies (${f}_{{\tau}_{Hz}}=$ 0.4 Hz; 0.15 Hz; 0.1 Hz; 0.04 Hz), computed for the four experimental conditions using eVAR (first row) and eVARFI (second row) identification methods. Statistically significant differences between pairs of conditions are marked with an asterisk.

**Figure 4.**Distribution of the univariate (

**a**), bivariate (

**b**) with systolic arterial pressure (SAP) and (

**c**) with respiration (RESP), and multivariate (

**d**) complexity measures depicted as boxplot (mean and confidence intervals, yellow filled box; standard deviation, black vertical line) and original values (dots) for selected frequencies (ftHz = 0.4 Hz; 0.15 Hz; 0.1 Hz; 0.04 Hz), computed for the four experimental conditions using eVAR (first row) and eVARFI (second row) identification methods. Statistically significant differences between pairs of conditions are marked with an asterisk.

**Figure 5.**Distribution of the long-rang parameter ${d}_{i}$ for each of the time series considered (heart period (HP), SAP and RESP) and for the four conditions.

**Table 1.**Significant differences (p-value < 0.05) between pair of conditions for each measure and frequency. The arrows indicate if the measure increases or decreases from rest to stress.

Measure | Approach | SU${}_{1}$ → UP | SU${}_{2}$ → MA | ||||||
---|---|---|---|---|---|---|---|---|---|

0.4 | 0.15 | 0.1 | 0.04 | 0.4 | 0.15 | 0.1 | 0.04 | ||

${\overline{C}}_{\mathbf{X}}$ | eVAR | ↘ | ↗ | ↗ | ↗ | ||||

eVARFI | ↘ | ↘ | ↘ | ||||||

${\overline{C}}_{{X}_{HP}|{X}_{HP}}$ | eVAR | ↘ | |||||||

eVARFI | ↘ | ↘ | |||||||

${\overline{C}}_{{X}_{HP}|{X}_{SAP},{X}_{HP}}$ | eVAR | ↘ | ↘ | ||||||

eVARFI | ↘ | ↘ | ↘ | ||||||

${\overline{C}}_{{X}_{HP}|{X}_{RESP},{X}_{HP}}$ | eVAR | ↘ | ↘ | ||||||

eVARFI | ↘ | ||||||||

${\overline{C}}_{{X}_{HP}|\mathbf{X}}$ | eVAR | ↘ | ↘ | ||||||

eVARFI | ↘ | ↘ |

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**MDPI and ACS Style**

Martins, A.; Pernice, R.; Amado, C.; Rocha, A.P.; Silva, M.E.; Javorka, M.; Faes, L. Multivariate and Multiscale Complexity of Long-Range Correlated Cardiovascular and Respiratory Variability Series. *Entropy* **2020**, *22*, 315.
https://doi.org/10.3390/e22030315

**AMA Style**

Martins A, Pernice R, Amado C, Rocha AP, Silva ME, Javorka M, Faes L. Multivariate and Multiscale Complexity of Long-Range Correlated Cardiovascular and Respiratory Variability Series. *Entropy*. 2020; 22(3):315.
https://doi.org/10.3390/e22030315

**Chicago/Turabian Style**

Martins, Aurora, Riccardo Pernice, Celestino Amado, Ana Paula Rocha, Maria Eduarda Silva, Michal Javorka, and Luca Faes. 2020. "Multivariate and Multiscale Complexity of Long-Range Correlated Cardiovascular and Respiratory Variability Series" *Entropy* 22, no. 3: 315.
https://doi.org/10.3390/e22030315