# Complexity and Entropy in Physiological Signals (CEPS): Resonance Breathing Rate Assessed Using Measures of Fractal Dimension, Heart Rate Asymmetry and Permutation Entropy

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{*}

## Abstract

**:**

^{®}GUI (graphical user interface) providing multiple methods for the modification and analysis of physiological data. Methods: To demonstrate the functionality of the software, data were collected from 44 healthy adults for a study investigating the effects on vagal tone of breathing paced at five different rates, as well as self-paced and un-paced. Five-minute 15-s recordings were used. Results were also compared with those from shorter segments of the data. Electrocardiogram (ECG), electrodermal activity (EDA) and Respiration (RSP) data were recorded. Particular attention was paid to COVID risk mitigation, and to parameter tuning for the CEPS measures. For comparison, data were processed using Kubios HRV, RR-APET and DynamicalSystems.jl software. We also compared findings for ECG RR interval (RRi) data resampled at 4 Hz (4R) or 10 Hz (10R), and non-resampled (noR). In total, we used around 190–220 measures from CEPS at various scales, depending on the analysis undertaken, with our investigation focused on three families of measures: 22 fractal dimension (FD) measures, 40 heart rate asymmetries or measures derived from Poincaré plots (HRA), and 8 measures based on permutation entropy (PE). Results: FDs for the RRi data differentiated strongly between breathing rates, whether data were resampled or not, increasing between 5 and 7 breaths per minute (BrPM). Largest effect sizes for RRi (4R and noR) differentiation between breathing rates were found for the PE-based measures. Measures that both differentiated well between breathing rates and were consistent across different RRi data lengths (1–5 min) included five PE-based (noR) and three FDs (4R). Of the top 12 measures with short-data values consistently within ± 5% of their values for the 5-min data, five were FDs, one was PE-based, and none were HRAs. Effect sizes were usually greater for CEPS measures than for those implemented in DynamicalSystems.jl. Conclusion: The updated CEPS software enables visualisation and analysis of multichannel physiological data using a variety of established and recently introduced complexity entropy measures. Although equal resampling is theoretically important for FD estimation, it appears that FD measures may also be usefully applied to non-resampled data.

## 1. Introduction

#### Objectives

- To conduct brief literature reviews on fractal dimension (FD) and HRA measures, and a more extensive review on resonance breathing.
- To use CEPS and DynamicalSystems.jl to analyse RRi, respiration and EDA data, and to compare results.
- To compare findings when using a variety of CEPS FD, HRA and measures based on permutation entropy (among others) to investigate whether there are marked differences between the effects of paced, self-paced and non-paced breathing on such physiological data—for example, which measures are most/least responsive to changes in breathing rate.
- To examine changes and agreement in key measures between baseline or self-paced breathing and optimal (or ‘resonance’) breathing, and explore questions such as ‘do people breathe naturally at their ideal rate?’
- To investigate the effects of parameter tuning on these measures in this context.
- To update the online CEPS ‘Primer’ and Manual to take changes in CEPS into account.
- To assess whether and which complexity and entropy measures applied to RRi and respiration data may be more effective at differentiating between resonance breathing and other breathing states than some of the more conventional HRV indices.
- To examine briefly whether age, sex, perceived stress (‘Distress’ and its converse, ‘Coping’), ‘Mindful awareness’ and two dimensions of interoceptive awareness (‘Noticing’, or awareness of body sensations, and ‘Attention regulation’, or the ability to sustain and control attention to body sensation), as well as a third dimension, ‘Self-Regulation’, may affect how CEPS measures reflect breathing state.
- To explore correlations within ‘families’ of measures, and between individual measures when applied to different data types (RRi, respiration and EDA).
- To investigate the effects of different data lengths on standard HRV and CEPS measures, with a view to determining the shortest data length that is feasible for use in further research on self-training methods of stress management.
- To explore how modifying the data in different ways (interpolation or deduplication, resampling, detrending, normalisation, multi-scaling, addition of noise) affects HRV and CEPS measures, and whether some of these methods may in fact compensate for the effects of shortening data length.
- In conclusion, to determine which measures are most useful for differentiating between resonance breathing and other breathing states, while also performing well for short data.

## 2. Materials and Methods

#### 2.1. Literature Reviews

#### 2.1.1. Fractal Dimension (FD) and Heart Rate Asymmetry (HRA) Measures

#### Fractal Dimension

_{2}). The latter, although it does provide a measure of FD, requires relatively long data samples for accurate estimation [28], so will not be considered further in this paper. A third review [29] includes several methods in addition to the box-count estimator, with code available in R [30]. A more recent and useful review of (mostly box-count) FDs, with code in Julia, is that by Datseris et al. (2021) [6], with the associated code available on GitHub [https://datseris.github.io/] (accessed on 20 January 2023).

#### Heart Rate Asymmetry (HRA)

#### 2.1.2. Resonance Breathing and Vagally-Mediated Heart Rate Variability (vmHRV)

#### 2.2. Study Protocol

#### 2.2.1. Resonant Breathing Rate Selection Using Paced Breathing

^{2}(LFBP); (2) power of the LF Spectral peak, in units of ms

^{2}(peak low frequency power, or PLFP); (3) normalised LF HRV, in normalised units (nu); (4) peak-to-trough difference in heart rate (HR), or ‘HRMaxMin’ (in beats per minute); (5) Phase Relationship of HR to Respiration rate (in degrees); and (6) comfort level. Shaffer and Meehan [57] also discussed how to select the RBR and how to ‘Break Ties’ when different breathing rates score maximally on different measures. This does, however, require a degree of clinical judgement.

#### 2.2.2. Ethics

#### COVID Risk Mitigation

#### 2.2.3. Participants

#### 2.2.4. Data Collection

#### 2.2.5. Software and Data Processing

#### Updating CEPS for This Project

#### Comparison with Estimators from DynamicalSystems.jl

_{2}, fractal dimension estimators from DynamicalSystems.jl and two timeseries complexity estimators. The fractal dimension Δ is fundamentally different from the time series FDs considered so far in this paper. Instead of quantifying the ‘roughness’ of the graph of a function (like the Higuchi estimator), Δ quantifies the effective dimensionality of the underlying dynamics. Specifically, we first analyse each time series using the approach of [118] to estimate an optimal delay embedding that most accurately represents the underlying dynamical attractor representing the dynamics generating the data. Once that is estimated, we reconstruct the attractors by delay embedding the time series. On this higher-dimensional object we use the well-established Grassberger-Procaccia algorithm [119] to estimate a fractal dimension as the scaling of the correlation sum versus a size parameter. Notice that while typically the reconstructed attractor would be higher than 2-dimensional, here we purposefully only embed up to two dimensions, to force the fractal dimension into the interval (1, 2), as used for the Higuchi dimension (to enable a simpler numerical comparison across the two methods). Unbounded values of Δ were also computed, and two optimal delay times, tau and tau2. In DynamicalSystems.jl, the Higuchi dimension was computed using values of k from 2 to 256, exponentially spaced, and the resulting values averaged. The other two complexity measures we used from DynamicalSystems.jl are wavelet entropy (‘wavent’) [117] and permutation entropy (‘perment’, or PE, with order m = 3 or 4 and lag as either 1 or the least mutual similarity time of the timeseries) [120]. Both measures were chosen because they are suitable quantifiers of complexity of timeseries and useful in classification tasks (such as the ones we attempt here), but also because they have been shown to be effective even with very short or non-stationary time series lengths (which we also have here). FD_H was also computed.

#### Other Software Used

^{®}Excel

^{®}2019.

#### 2.2.6. Data Processing

#### 2.2.7. Data Pre-Processing and Modification

#### Detrending

#### Data Segmentation (‘Cut Files’)

#### Adding Noise (‘Add Noise’)

#### Interpolation

#### Equal Resampling, Using ‘Shape-Preserving Piecewise Cubic Spline Interpolation’

#### 2.2.8. Parameter Selection

_{max}for FD_H based on a very small sample (N = 9) and found that the measure differentiated reasonably well between normal and paced breathing with k

_{max}= 5, or for k

_{max}between 9 and 14. Here, we went on to use several strategies to tune parameters:

_{max}that no longer showed a significant difference between the Baseline and RBR Trials, as would have been expected from our previous study.

#### 2.2.9. Statistical Analysis

#### Data Distribution

#### Analysis of Variance 1. Welch’s ANOVA

#### Analysis of Variance 2. Friedman Tests, Kendall’s W and Conover Tests

#### Assessing Agreement. Intraclass Correlation Coefficients (ICCs) and Simple Correlations

#### Combining the Results of Conover Tests and ICCs

- CEPS and RR-APET measures for non-resampled RRi data;
- CEPS and DynamicalSystems.jl measures for RRi data resampled at 4 Hz;
- CEPS measures for RRi data resampled at 10 Hz;
- CEPS and DynamicalSystems.jl measures for detrended and deduplicated EDA data.

#### Effects of Age, Sex, Perceived Stress and Other Trait and State Measures

#### Correlations within ‘Families’ of Measures, and between Individual Measures

## 3. Results

#### 3.1. Normality of Data

#### 3.2. Data Resampling and Modification

#### 3.2.1. The Effects of Data Resampling on CEPS Measures

#### 3.2.2. The Effects of Data Modification—Mitigating for the Effects of Data Segmentation (Shortening)

#### 3.3. Parameter Tuning

#### 3.4. CEPS, DynamicalSystems.jl and Other Analysis of RRi, Respiration and EDA Data

^{7}, based on time series length. In CEPS, on the other hand, k values were from k = 1 to k

_{max}= 2 to 15 (i.e., linear spaced values up to a varied k

_{max}, choosing the k

_{max}with best discriminatory power for our application); (2) different line-fitting functions were used in DynamicalSystems.jl (fitting a slope to an identified linear scaling region as described in Datseris et al. 2021 [6]) and CEPS (standard MATLAB ‘polyfit’ polynomial curve fitting).

#### 3.4.1. Five-Minute ECG RRi Data—CEPS, DynamicalSystems.jl and Kubios HRV Analysis

^{2}and Kendall’s W were used, as described in Section 2.2.9). Medians are shown in Table 6, with interquartile ranges (IQRs) in parentheses.

^{2}and W were greater for the permutation entropy family of measures than for the others, except for the RRi data resampled at 10 Hz, for which χ

^{2}and W were greatest for the HRA family of measures.

_{2}, if slightly greater for the former.

^{2}> 150) are shown in Table 7, with results for the best-performing 4R measures from Kubios HRV provided as a comparison (see [138] for details).

^{2}< 10). For the RRi data resampled at both 4 Hz and10 Hz, these were FD_Moisy_Box (FD), LZC (OC), SlopeEn (OE), and two RQA measures (RTmax and Lmax); for the data resampled at 4 Hz, they also included SI (HRA) and AAPE (PE), with GridEn (OE) for the data resampled at 10 Hz. For the un-resampled data, measures were EPP SD2_6 (HRA), CAFE (OE) and three RQA measures (including Lmax once again). Of the Kubios HRV measures, HFpwr (AR) also appeared to be little affected by respiration rate.

#### Post-Hoc Analysis

#### 3.4.2. Respiration Data—CEPS Analysis Only

^{2}> 150 are shown in Table 9. Note that maximal χ

^{2}values are lower than for the RRi data.

^{2}. The measure MmSE13 is not included in this Figure, as although it was highest at baseline it remained unchanged for all paced breathing rates.

#### 3.4.3. EDA Data—CEPS and DynamicalSystems.jl Analysis

^{2}> 150. For the following measures, Friedman’s χ

^{2}was greater than 28: RMSSD (χ

^{2}= 29.035), EPP SD1_1 to SD1_-7 excluding SD1_6 (χ

^{2}= 28.303–28.824), and FD_K (χ

^{2}= 28.767).

#### 3.4.4. Summary of Results for RRi, Respiration and EDA Data

^{2}and W are lower for the top two DynamicalSystems.jl measures than for the corresponding CEPS measures. Only wavelet entropy (Wavent) shows a reasonable effect size (Kendall’s W > 0.4). Patterns of change with breathing frequency differ from those observed for the corresponding CEPS measures (compare Figure 5, Figure 6 and Figure 7 above with Figure 8 below).

#### 3.4.5. Some Findings on Heart Rate Asymmetry (HRA)

_{up}and SDNN

_{down}performed best (both with S > 12), followed by SD2

_{up}(S = 11.982). Values of S > 10 were also obtained for PI, GI and SD1down. By way of comparison, FD measures such as mFD_M, FD_PRI and FD_H all resulted in values of S > 14. For 10 out of the 16 HRA measures analysed here, Conover’s S was lower for the RRi data resampled at 4 Hz.

#### Correlations between HRA Indices and HRV Measures

#### Respiration and Asymmetry

#### 3.4.6. Difference and Agreement between Baseline or Self-Paced Breathing and Optimal (or ‘Resonance’) Breathing or Breathing at 5 BrPM: Do Measure Values during Slow Self-Paced Breathing Predict Those of RBR?

#### 3.4.7. Results for Correlations within ‘Families’ of Measures, and between Individual Measures When Applied to Different Data Types (RRi, Respiration and EDA) Are Described in the Supplementary Materials (Section SM5.1)

#### 3.5. The Effects of Time

#### 3.5.1. Data Length and Its Effect on Different Measures

#### Data Length and Differences in Measures between Breathing Rates

^{2}and Kendall’s W at 5 min, but FD_C and FD_H providing more consistently similar values across all data lengths (median W 0.371, IQR 0.354 to 0.720, and median W 0.632, IQR 0.599 to 0.633, respectively).

^{2}, Kendall’s W and the Conover statistic for the Baseline to RBR differences were calculated for each measure in the three main families (FDs, HRAs and PE-based measures). Measures were considered if they showed standardised median values of Friedman’s χ

^{2}and Kendall’s W > 0.8, or of the Conover statistic S for differences between Baseline and the composite RBR trial. Numerically, these thresholds were approximately 150, 0.480 and 12, respectively, for the non-resampled RRi data, 145, 0.466 and 10.5 for the data resampled at 4 Hz, and 20, 0.070 and 3.3 for the EDA data. The corresponding thresholds for RR-APET were 100, 0.319 and 9.4. The EDA thresholds were low, unlikely to be useful in practice, so were not examined further here, nor were any measures with values lower than any one of these thresholds for one or more of the 2-, 3-, 4- or 5-min data segments. This draconian limitation reduced the number of measures that might be serviceable for analysis of short data and for differentiation between trials to something manageable, although excluding many measures that might otherwise have been useful for one or the other, particularly for the RRi (4R) data. Results are shown in Table 16.

#### Agreements between Measures for Different Data Lengths

_{down}and SDNN

_{down}), with wavelet entropy providing the best ICC (0.991) of the RRi (4R) measures, although a low S (0.762), and FD_PRI providing a high S (0.966) but a low ICC (0.599). For the EDA data, ICC was >0.999 for ‘Jitta’ and 0.966 for FD_K, although S was not > 0.8 for either of these measures.

#### 3.5.2. Do Nonlinear Measures Indicate RBR More Accurately than Standard HRV Measures, Especially for Short Data?

_{a}achieved effect sizes > 0.2.

_{a}achieved effect sizes > 0.2, as for the RRi (noR) data.

_{max}were sometimes >0.25, but were also very variable over the different durations, while for LF power the effect size only exceeded 0.25 for the 5-min data.

## 4. Discussion

#### 4.1. General Points

#### 4.2. Our Basic Approach

^{2}, with Kendall’s W as a measure of effect size) and Conover tests show more significant differences for the nonlinear than the standard HRV measures, and if these differences also hold for shortened data, then further use of carefully selected nonlinear measures from those available in CEPS can be justified. However, caution should still be exercised when interpreting results from the Conover tests for non-resampled data, in that variance may be greater than when data have not been equally resampled. Pragmatically, we have found FD_H to be a useful measure for differentiating the effects of breathing rate on non-resampled RRi data. On the other hand, FD_H, as well as several other FD measures, did not result in useful values of Friedman’s χ

^{2}for either 5-min or shorter RRi (4R) data, whereas they did for the non-resampled RRi data, and the clear differences for the two data types in Figure 13 reinforce the importance of careful selection of the type of data to use and careful interpretation of results. For RRi data, for example, measures which appeared less affected by respiration rate included some of the RQA, Jitter (frequency variation from cycle to cycle [101]), LLE, EPP r measures and LZC. For the other data types, different groupings of measures were unresponsive to respiration rate, but there was no obvious pattern to these.

#### 4.3. The Anxieties of Data Collection and Collaboration

#### 4.4. Including EDA Results

#### 4.5. An Explanation of HRA Results

#### 4.6. Limitations

#### 4.7. Advantages

## 5. Conclusions and Future Directions

#### 5.1. Conclusions

#### 5.2. Future Directions

## Supplementary Materials

**[mdpi to complete]**:

**SM1.**Parameter Tuning,

**SM2.**Changes in CEPS measures over time,

**SM3.**Effects of age, sex, perceived stress and other trait and state measures on CEPS and Kubios HRV measures,

**SM4.**Respiration and Asymmetry,

**SM5**. Some findings on correlation. CEPS itself, together with A primer on Complexity and Entropy and a User Manual, are available online at https://github.com/harikalakandel/CEPSv2/tree/master (accessed on 20 January 2023).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

4R, 10R | Data resampled at 4 Hz or 10 Hz |

AAPE | Amplitude-aware Permutation Entropy |

ACR5 | Autocorrelation at lag 5 |

AE | Average entropy |

AI | Area Index |

Alpha1 | See DFA Alpha1 |

Alpha2 | See DFA Alpha2 |

ApEn | Approximate entropy |

AR | Autoregressive |

ASI | Asymmetric Spread Index |

AttnEn | Attention entropy |

AvgApEnP | Average Approximate entropy based on profiling |

AvgSampEnP | Average Sample entropy based on profiling |

B_ApEn | Bucket-assisted Approximate entropy |

B_SampEn | Bucket-assisted Sample entropy |

BBi | Breath-to-Breath interval |

BE | Bubble Entropy |

BrPM | Breaths Per Minute |

C0 | C0 complexity, a representation of sequence randomness |

C1_{a} | Relative contribution of accelerations to short-term variance in HRA |

C1_{d} | Relative contribution of decelerations to short-term variance in HRA |

C2_{a} | Relative contribution of accelerations to long-term variance in HRA |

C2_{d} | Relative contribution of decelerations to long-term variance in HRA |

CAFE | Centred and averaged fuzzy entropy |

CCM | Complex Correlation Measure |

CEPS | Complexity and Entropy in Physiological Signals |

χ^{2} | “Chi-square” statistic from the non-parametric Friedman test |

CI | Complexity index |

CID | Complexity-invariant distance |

CmSE | Composite multiscale entropy |

COPD | Chronic Obstructive Pulmonary Disease |

CoSEn | Coefficient of Sample entropy |

CoSiEn | Cosine Similarity Entropy |

CPEI | Composite permutation entropy index |

CV | Coefficient of Variation |

CVs | Coefficients of Variation |

D_{2} | Correlation Dimension |

DE | Diffusion entropy |

Dedup | Deduplicated |

Δ and Δ2 | Fractal dimension estimators in DynamicalSystems.jl |

DFA | Detrended Fluctuation Analysis |

DFA Alpha 1 | Detrended Fluctuation Analysis short-term scaling exponent |

DFA Alpha 2 | Detrended Fluctuation Analysis long-term scaling exponent |

DiffEn | Differential entropy |

DistEn | Distribution Entropy |

DS | DynamicalSystems.jl |

Dβ | Spectral dimension |

Dσ | Variance dimension |

ECG | Electrocardiogram |

ECG IBI | ECG Interbeat interval |

EDA | Electrodermal activity |

EE | Energy entropy |

EI | Ehlers’ Index |

EoD | Entropy of difference |

EPE | Edge Permutation Entropy |

EPP | Extended Poincaré Plot |

EPP r1 | Pearson’s r at lag 1 in the Extended Poincaré Plot |

EPP SD1_2 | SD1 at lag 2 in the Extended Poincaré Plot |

ES | Effect size |

ESCHA | Emergence, Self-organization, Complexity, Homeostasis and Autopoiesis (here, only Complexity has been used) |

ESCHA_c | ESCHA for continuous data |

ESCHA_d | ESCHA for discrete data |

FD | Fractal Dimension |

FD_Amp | Amplitude fractal dimension |

FD_Box_Moisy | Box-counting fractal dimension, using Moisy’s implementation |

FD_Box_MvdL | Box-counting fractal dimension, according to Meerwijk and van der Linden |

FD_C | Castiglioni fractal dimension |

FD_Dist | Distance fractal dimension |

FD_H | Higuchi fractal dimension (for which we used ‘HFD’ in our earlier paper [4]) |

FD_K | Katz fractal dimension |

FD_LRI | Fractal dimension based on linear regression intersection |

FD_M | Mandelbrot fractal dimension |

FD_P | Petrosian fractal dimension |

FD_PRI | Fractal dimension based on polynomial regression intersection |

FD_S | Sevcik fractal dimension |

FD_Sign | Sign fractal dimension |

FFT | Fast Fourier transform |

GI | Guzik’s index |

GPP | Generalised Poincaré Plot |

GridEn | Grid Entropy (or Gridded Distribution entropy) |

GUI | Graphical user interface |

HF | High frequency |

HFpwr | (Lomb-Scargle)High frequency power, based on the Lomb-Scargle periodogram |

HFpwr (Welch) | High frequency power, based on the Welch periodogram |

HR | Heart rate |

HRA | Heart Rate Asymmetry |

HRMaxMin | Peak-to-trough difference in heart rate |

HRV | Heart Rate Variability |

Hz | Hertz (unit of frequency) |

ICC | Intraclass Correlation Coefficient |

ImPE | Improved multiscale Permutation Entropy |

INbreath | Inbreath data |

IncrEn | Increment entropy |

IQR | interquartile range |

Jitter_Jitt | Local jitter, or average absolute difference in length between two consecutive periods, divided by average period |

Jitter_Jitta | Absolute jitter, or average absolute difference in length between two consecutive periods |

Jitter_ppq5 | Average absolute difference between a period and the average of it and the two previous and two subsequent periods, divided by the average period |

Jitter-RAP | Relative Absolute Perturbation, or average absolute difference between a period and the average of it and its two neighbours, divided by the average period |

KLD | Kullbach-Leibler Divergence |

k_{max} | Maximum interval time used in calculation of FD_H |

L_ApEn | Lightweight Approximate entropy |

L_SampEn | Lightweight Sample entropy |

LF | Low frequency |

LFBP | Low frequency band power |

LFpwr | Low frequency power |

LLE32 | Largest Lyapunov exponent, iteration 32 |

LS | Lomb-Scargle |

LZC | Lempel-Ziv complexity |

LZPC | Lempel Ziv Permutation Complexity |

m | Order, or embedding dimension |

MAAS | Mindful Attention Awareness Scale |

MAIA | Multidimensional Assessment of Interoceptive Awareness |

MESA | Maximum Entropy Spectral Analysis |

mFD_M | multiscale fractal dimension, according to Maragos |

mFmDFA | multifractal multiscale detrended fluctuation analysis |

mLZC7 | multiscale Lempel-Ziv complexity, at Scale 7 |

MmSE | Modified multiscale Sample Entropy, at Scale indicated by number following abbreviation |

mPE | Multiscale Permutation entropy, at Scale indicated by number following abbreviation |

mPE1 | multiscale Permutation entropy 1 |

mPhEn | multiscale Phase entropy |

mPM_E | multiscale Permutation Min-entropy |

n or N | Number |

n.p. | Not published |

NLD | Normalised Length Density (fractal dimension according to Kalauzi) |

NLDiL_m | NLD based on normalisation of amplitudes for whole signal (mean, using Log model) |

NLDiL_sd | NLD based on normalisation of amplitudes for whole signal (standard deviation, using Log model) |

NLDiP_m | NLD fractal dimension based on normalisation of amplitudes for whole signal (mean, using Power model) |

NLDiP_sd | NLD based on normalisation of amplitudes for whole signal (standard deviation, using Power model) |

NLDwL_m | NLD based on normalisation of moving window amplitudes (mean, using Log model) |

NLDwL_sd | NLD based on normalisation of moving window amplitudes (standard deviation, using Log model) |

NLDwP_m | NLD based on normalisation of moving window amplitudes (mean, using Power model) |

NLDwP_sd | NLD based on normalisation of moving window amplitudes (standard deviation, using Power model) |

noR | Non-resampled |

nu | Normalised units |

OC | Family of ‘Other Complexity’ measures |

OE | Family of ‘Other Entropy’ measures |

ORDO | Open Research Data Online (Open University Repository) |

OU | Open University |

OUTbreath | Outbreath data |

PCR | Polymerase chain reaction |

PE | Family of measures based on ‘Permutation entropy’ |

PI | Porta’s index |

PJSC | Permutation Jensen-Shannon complexity |

PLFP | Peak low frequency power |

PLZC | Permutation Lempel Ziv Complexity |

pNN50 | percentage of absolute differences in successive ECG NN values > 50 ms |

PNS | Parasympathetic nervous system index, from Kubios HRV |

PP | Peak-to-peak |

PPG | Photoplethysmography |

PSS | Perceived Stress Scale |

PTSD | post-traumatic stress disorder |

QR | Quick response |

QSE | Quadratic Sample entropy |

r1 and r2 | See EPP r1 and EPP r2 |

RBA | Resonant breathing assessment |

RBR | Resonance breathing rate |

RCmDE3 | Refined Composite multiscale Dispersion Entropy at lag 3 |

RE | Rényi entropy |

RespR | median Outbreath-to-Inbreath ratio |

rest | Breathing trials other than RBR |

RMSSD | Root mean square of successive differences between normal heartbeats |

RoCV | Robust Coefficient of Variation |

RoSlope | Robust Slope |

RPDE | Recurrence period density entropy |

RPE | Rényi Permutation Entropy |

RQA | Family of measures based on recurrence quantification analysis |

RQA DET | Recurrence Quantification Analysis: Determinism |

RQA Lmax | Recurrence Quantification Analysis: Max diagonal line length |

RQA Lmean | Recurrence Quantification Analysis: Mean diagonal line length |

RQA RTmax | Recurrence Quantification Analysis: Max recurrence time |

RQA Vmax | Recurrence Quantification Analysis: Max vertical line length |

RQA Vmean | Recurrence Quantification Analysis: Mean vertical line length |

RR-APET | Python-based Heart rate variability analysis software |

RRi | ECG RR interval |

RSA | Respiratory sinus arrhythmia |

RSP | Respiration |

SampEn | Sample entropy |

SD | Standard Deviation |

SD1 | Standard Deviation of Poincaré Plot scattergram (minor axis) |

SD1_2 | See EPP SD1_2 |

SD1_{down} | SD1 for the number of points below the Poincaré Plot line of identity |

SD1_{up} | SD1 for the number of points above the Poincaré Plot line of identity |

SD2 | Standard Deviation of Poincaré Plot scattergram (major axis) |

SD2_{down} | SD2 for the number of points below the Poincaré Plot line of identity |

SD2_{up} | SD2 for the number of points above the Poincaré Plot line of identity |

SDNN | Standard deviation of the interbeat intervals of normal sinus beats |

SDNN_{down} | Deceleration-related part of HRV measure SDNN (Standard Deviation of interbeat interval of normal sinus beats) |

SDNN_{up} | Acceleration-related part of HRV measure SDNN (Standard Deviation of interbeat interval of normal sinus beats) |

Shimmer_apq3 | Average absolute difference between amplitude of a period and the mean amplitudes of its two neighbours, divided by the average amplitude |

Shimmer_apq5 | Average absolute difference between amplitude of a period and the mean amplitudes of it and its four nearest neighbours, divided by the average amplitude |

Shimmer_ShdB | Average absolute difference of base 10 logarithm of the amplitude difference between two consecutive periods |

Shimmer_Shim | Average absolute difference between amplitudes of two consecutive periods, divided by the average amplitude |

SI | Slope index |

SlopeEn | Slope entropy |

SNS | Sympathetic nervous system index, from Kubios HRV |

SpEn | Spectral entropy |

SPSS | Statistical Package for Social Science |

SQA | Symmetry Quantification Analysis |

SymDyn | Symbolic Dynamics |

Tangle | Temporal complexity metric |

TE | Tsallis entropy |

T_E | Tone_entropy (either T_E Tone or T_E Entropy) |

Totpwr | Total power |

TPE | Tsallis Permutation Entropy |

UCFB | University Campus of Football Business |

VM | Volatility Method |

vmHRV | Vagally mediated HRV |

vmHRVBF | Vagally-mediated heart rate variability biofeedback |

W | Kendall’s coefficient of concordance |

wavent (or WE) | Wavelet entropy |

## References

- Li, P. EZ Entropy: A Software Application for the Entropy Analysis of Physiological Time-Series. Biomed. Eng. Online
**2019**, 18, 1–15. [Google Scholar] [CrossRef] [PubMed] - Azami, H.; Faes, L.; Escudero, J.; Humeau-Heurtier, A.; Silva, L.E.V. Entropy Analysis of Univariate Biomedical Signals: Review and Comparison of Methods. In Frontiers in Entropy across the Disciplines: Panorama of Entropy: Theory, Computation, and Applications; World Scientific Publishing: Singapore, 2020; pp. 233–286. [Google Scholar] [CrossRef]
- Humeau-Heurtier, A. Entropy Analysis in Health Informatics. In Intelligent Systems Reference Library; Springer Science and Business Media: Berlin/Heidelberg, Germany, 2021; Volume 192, pp. 123–143. [Google Scholar] [CrossRef]
- Mayor, D.; Panday, D.; Kandel, H.K.; Steffert, T.; Banks, D. CEPS: An Open Access Matlab Graphical User Interface (GUI) for the Analysis of Complexity and Entropy in Physiological Signals. Entropy
**2021**, 23, 321. [Google Scholar] [CrossRef] [PubMed] - Flood, M.W.; Grimm, B. EntropyHub: An Open-Source Toolkit for Entropic Time Series Analysis. PLoS ONE
**2021**, 16, e0259448. [Google Scholar] [CrossRef] [PubMed] - Datseris, G.; Kottlarz, I.; Braun, A.P.; Parlitz, U. Estimating the Fractal Dimension: A Comparative Review and Open Source Implementations. arXiv
**2021**, arXiv:2109.05937. [Google Scholar] [CrossRef] - Datseris, G. DynamicalSystems.Jl: A Julia Software Library for Chaos and Nonlinear Dynamics. J. Open Source Softw.
**2018**, 3, 598. [Google Scholar] [CrossRef] - Kalauzi, A.; Bojić, T.; Rakić, L. Extracting Complexity Waveforms from One-Dimensional Signals. Nonlinear Biomed. Phys.
**2009**, 3, 8. [Google Scholar] [CrossRef] - Platiša, M.M.; Radovanović, N.N.; Kalauzi, A.; Pavlović, S. Generalized Poincaré Plots Analysis of Cardiac Interbeat Intervals in Heart Failure. In Proceedings of the 15th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2022), Virtual, 9–11 February 2022; pp. 251–256. [Google Scholar] [CrossRef]
- Grigolini, P.; Palatella, L.; Raffaelli, G. Asymmetric Anomalous Diffusion: An Efficient Way to Detect Memory in Time Series. Fractals
**2001**, 9, 439–449. [Google Scholar] [CrossRef] - Jelinek, H.F.; Tuladhar, R.; Culbreth, G.; Bohara, G.; Cornforth, D.; West, B.J.; Grigolini, P. Diffusion Entropy vs. Multiscale and Rényi Entropy to Detect Progression of Autonomic Neuropathy. Front. Physiol.
**2021**, 11, 607324. [Google Scholar] [CrossRef] - Ponce-Flores, M.; Frausto-Solís, J.; Santamaría-Bonfil, G.; Pérez-Ortega, J.; González-Barbosa, J.J. Time Series Complexities and Their Relationship to Forecasting Performance. Entropy
**2020**, 22, 89. [Google Scholar] [CrossRef] - Harte, D. Multifractals: Theory and Applications; Chapman and Hall/CRC: Boca Raton, FL, USA, 2001. [Google Scholar] [CrossRef]
- Higuchi, T. Approach to an Irregular Time Series on the Basis of the Fractal Theory. Phys. D Nonlinear Phenom.
**1988**, 31, 277–283. [Google Scholar] [CrossRef] - Katz, M.J. Fractals and the Analysis of Waveforms. Comput. Biol. Med.
**1988**, 18, 145–156. [Google Scholar] [CrossRef] - Castiglioni, P. What Is Wrong in Katz’s Method? Comments on: “A Note on Fractal Dimensions of Biomedical Waveforms”. Comput. Biol. Med.
**2010**, 40, 950–952. [Google Scholar] [CrossRef] - Petrosian, A. Kolmogorov Complexity of Finite Sequences and Recognition of Different Preictal EEG Patterns. In Proceedings of the IEEE Symposium on Computer-Based Medical Systems, Lubbock, TX, USA, 9–10 June 1995; pp. 212–217. [Google Scholar] [CrossRef]
- Sevcik, C. A Procedure to Estimate the Fractal Dimension of Waveforms. Complex. Int.
**1998**, 5. [Google Scholar] - Moisy, F. Boxcount. Available online: https://uk.mathworks.com/matlabcentral/fileexchange/13063-boxcount (accessed on 23 July 2022).
- Meerwijk, E.L.; Ford, J.M.; Weiss, S.J. Resting-State EEG Delta Power Is Associated with Psychological Pain in Adults with a History of Depression. Biol. Psychol.
**2015**, 105, 106–114. [Google Scholar] [CrossRef] - Kizlaitienė, I. Fractal Modeling of Speech Signals. Master’s Thesis, Vilnius Universitetas, Vilnius, Lithuania, 2021. Available online: https://epublications.vu.lt/object/elaba:81590289/81590289.pdf (accessed on 20 January 2023).
- Maragos, P. Fractal Signal Analysis Using Mathematical Morphology. Adv. Electron. Electron Phys.
**1994**, 88, 199–246. [Google Scholar] [CrossRef] - Zlatintsi, A.; Maragos, P. Multiscale Fractal Analysis of Musical Instrument Signals with Application to Recognition. IEEE Trans. Audio Speech Lang. Process.
**2013**, 21, 737–748. [Google Scholar] [CrossRef] - Kinsner, W. A Unified Approach to Fractal Dimensions. J. Inf. Technol. Res.
**2008**, 1, 62–85. [Google Scholar] [CrossRef] - Wen, T.; Cheong, K.H. The Fractal Dimension of Complex Networks: A Review. Inf. Fusion
**2021**, 73, 87–102. [Google Scholar] [CrossRef] - Henriques, T.; Ribeiro, M.; Teixeira, A.; Castro, L.; Antunes, L.; Costa-Santos, C. Nonlinear Methods Most Applied to Heart-Rate Time Series: A Review. Entropy
**2020**, 22, 309. [Google Scholar] [CrossRef] - Barabási, A.L.; Stanley, H.E. Fractal Concepts in Surface Growth. Z. Für Phys. Chem.
**1995**, 193, 218–219. [Google Scholar] [CrossRef] - Grassberger, P.; Procaccia, I. Characterization of Strange Attractors. Phys. Rev. Lett.
**1983**, 50, 346–349. [Google Scholar] [CrossRef] - Gneiting, T.; Ševčíková, H.; Percival, D.B. Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data. Stat. Sci.
**2012**, 27, 247–277. [Google Scholar] [CrossRef] - Ševčíková, H.; Percival, D.; Gneiting, T. Estimation of Fractal Dimensions: Package ‘Fractaldim’. Available online: https://CRAN.R-project.org/package=fractaldim (accessed on 20 January 2023).
- Mieszkowski, D.; Kośmider, M.; Krauze, T.; Guzik, P.; Piskorski, J. Asymmetric Detrended Fluctuation Analysis Reveals Asymmetry in the RR Intervals Time Series. J. Appl. Math. Comput. Mech.
**2016**, 15, 99–106. [Google Scholar] [CrossRef] - Martínez, J.H.; Herrera-Diestra, J.L.; Chavez, M. Detection of Time Reversibility in Time Series by Ordinal Patterns Analysis. Chaos
**2018**, 28, 123111. [Google Scholar] [CrossRef] [PubMed] - Czippelova, B.; Chladekova, L.; Uhrikova, Z.; Zibolen, M.; Javorka, K.; Javorka, M. Is the Time Irreversibility of Heart Rate Present Even in Newborns? In Proceedings of the 2014 8th Conference of the European Study Group on Cardiovascular Oscillations, ESGCO 2014, Trento, Italy , 25–28 May 2014; pp. 15–16. [Google Scholar] [CrossRef]
- Karmakar, C.K.; Khandoker, A.H.; Gubbi, J.; Palaniswami, M. Complex Correlation Measure: A Novel Descriptor for Poincaré Plot. Biomed. Eng. Online
**2009**, 8, 17. [Google Scholar] [CrossRef] [Green Version] - Ehlers, C.L.; Havstad, J.; Prichard, D.; Theiler, J. Low Doses of Ethanol Reduce Evidence for Nonlinear Structure in Brain Activity. J. Neurosci.
**1998**, 18, 7474–7486. [Google Scholar] [CrossRef] [PubMed] - Guzik, P.; Piskorski, J.; Krauze, T.; Wykretowicz, A.; Wysocki, H. Heart Rate Asymmetry by Poincaré Plots of RR Intervals. Biomed. Tech.
**2006**, 51, 272–275. [Google Scholar] [CrossRef] - Piskorski, J.; Guzik, P. Geometry of the Poincaré Plot of RR Intervals and Its Asymmetry in Healthy Adults. Physiol. Meas.
**2007**, 28, 287–300. [Google Scholar] [CrossRef] - Porta, A.; Guzzetti, S.; Montano, N.; Gnecchi-Ruscone, T.; Furlan, R.; Malliani, A. Time Reversibility in Short-Term Heart Period Variability. In Proceedings of the Computers in Cardiology Conference, Valencia, Spain, 17–20 September 2006; Volume 33, pp. 77–80. [Google Scholar]
- Karmakar, C.; Khandoker, A.; Palaniswami, M. Analysis of Slope Based Heart Rate Asymmetry Using Poincaré Plots. In Proceedings of the Computing in Cardiology Conference, Krakow, Poland, 9–12 September 2012; Volume 39, pp. 949–952. [Google Scholar]
- Yan, C.; Li, P.; Ji, L.; Yao, L.; Karmakar, C.; Liu, C. Area Asymmetry of Heart Rate Variability Signal. Biomed. Eng. Online
**2017**, 16, 112. [Google Scholar] [CrossRef] - Karmakar, C.K.; Khandoker, A.H.; Gubbi, J.; Palaniswami, M. Defining Asymmetry in Heart Rate Variability Signals Using a Poincaré Plot. Physiol. Meas.
**2009**, 30, 1227–1240. [Google Scholar] [CrossRef] - Rohila, A.; Sharma, A. Asymmetric Spread of Heart Rate Variability. Biomed. Signal Process. Control
**2020**, 60, 101985. [Google Scholar] [CrossRef] - Chladekova, L.; Czippelova, B.; Turianikova, Z.; Tonhajzerova, I.; Calkovska, A.; Baumert, M.; Javorka, M. Multiscale Time Irreversibility of Heart Rate and Blood Pressure Variability during Orthostasis. Physiol. Meas.
**2012**, 33, 1747–1756. [Google Scholar] [CrossRef] - Czippelova, B.; Chladekova, L.; Uhrikova, Z.; Javorka, K.; Zibolen, M.; Javorka, M. Time Irreversibility of Heart Rate Oscillations in Newborns—Does It Reflect System Nonlinearity? Biomed. Signal Process. Control
**2015**, 19, 85–88. [Google Scholar] [CrossRef] - Goshvarpour, A.; Goshvarpour, A. Asymmetry of Lagged Poincare Plot in Heart Rate Signals during Meditation. J. Tradit. Complement. Med.
**2021**, 11, 16–21. [Google Scholar] [CrossRef] - Alvarez-Ramirez, J.; Echeverria, J.C.; Meraz, M.; Rodriguez, E. Asymmetric Acceleration/Deceleration Dynamics in Heart Rate Variability. Phys. A Stat. Mech. Appl.
**2017**, 479, 213–224. [Google Scholar] [CrossRef] - Piskorski, J.; Kosmider, M.; Mieszkowski, D.; Krauze, T.; Wykretowicz, A.; Guzik, P. Properties of Asymmetric Detrended Fluctuation Analysis in the Time Series of RR Intervals. Phys. A Stat. Mech. Appl.
**2018**, 491, 347–360. [Google Scholar] [CrossRef] - Piskorski, J.; Guzik, P. Compensatory Properties of Heart Rate Asymmetry. J. Electrocardiol.
**2012**, 45, 220–224. [Google Scholar] [CrossRef] [PubMed] - Kurosaka, C.; Maruyama, T.; Yamada, S.; Hachiya, Y.; Ueta, Y.; Higashi, T. Estimating Core Body Temperature Using Electrocardiogram Signals. PLoS ONE
**2022**, 17, e0270626. [Google Scholar] [CrossRef] [PubMed] - Porges, S.W. A Phylogenetic Journey through the Vague and Ambiguous Xth Cranial Nerve: A Commentary on Contemporary Heart Rate Variability Research. Biol. Psychol.
**2007**, 74, 301–307. [Google Scholar] [CrossRef] - Jung, W.; Jang, K.I.; Lee, S.H. Heart and Brain Interaction of Psychiatric Illness: A Review Focused on Heart Rate Variability, Cognitive Function, and Quantitative Electroencephalography. Clin. Psychopharmacol. Neurosci.
**2019**, 17, 459–474. [Google Scholar] [CrossRef] [PubMed] - Appelhans, B.M.; Luecken, L.J. Heart Rate Variability as an Index of Regulated Emotional Responding. Rev. Gen. Psychol.
**2006**, 10, 229–240. [Google Scholar] [CrossRef] - Thayer, J.F.; Åhs, F.; Fredrikson, M.; Sollers, J.J.; Wager, T.D. A Meta-Analysis of Heart Rate Variability and Neuroimaging Studies: Implications for Heart Rate Variability as a Marker of Stress and Health. Neurosci. Biobehav. Rev.
**2012**, 36, 747–756. [Google Scholar] [CrossRef] [PubMed] - Ask, T.F.; Lugo, R.G.; Sütterlin, S. The Neuro-Immuno-Senescence Integrative Model (NISIM) on the Negative Association between Parasympathetic Activity and Cellular Senescence. Front. Neurosci.
**2018**, 12, 726. [Google Scholar] [CrossRef] - Vaschillo, E.; Lehrer, P.M.; Rishe, N.; Konstantinov, M. Heart Rate Variability Biofeedback as a Method for Assessing Baroreflex Function: A Preliminary Study of Resonance in the Cardiovascular System. Appl. Psychophysiol. Biofeedback
**2002**, 27, 1–27. [Google Scholar] [CrossRef] - Vaschillo, E.G.; Vaschillo, B.; Pandina, R.J.; Bates, M.E. Resonances in the Cardiovascular System Caused by Rhythmical Muscle Tension. Psychophysiology
**2011**, 48, 927–936. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shaffer, F.; Meehan, Z.M. A Practical Guide to Resonance Frequency Assessment for Heart Rate Variability Biofeedback. Front. Neurosci.
**2020**, 14, 1055. [Google Scholar] [CrossRef] - Lehrer, P.M.; Vaschillo, E.; Vaschillo, B.; Lu, S.E.; Eckberg, D.L.; Edelberg, R.; Shih, W.J.; Lin, Y.; Kuusela, T.A.; Tahvanainen, K.U.O.; et al. Heart Rate Variability Biofeedback Increases Baroreflex Gain and Peak Expiratory Flow. Psychosom. Med.
**2003**, 65, 796–805. [Google Scholar] [CrossRef] - Lehrer, P.M.; Vaschillo, E.; Vaschillo, B.; Lu, S.E.; Scardella, A.; Siddique, M.; Habib, R.H. Biofeedback Treatment for Asthma. Chest
**2004**, 126, 352–361. [Google Scholar] [CrossRef] - Lehrer, P.; Vaschillo, E.; Lu, S.E.; Eckberg, D.; Vaschillo, B.; Scardella, A.; Habib, R. Heart Rate Variability Biofeedback: Effects of Age on Heart Rate Variability, Baroreflex Gain, and Asthma. Chest
**2006**, 129, 278–284. [Google Scholar] [CrossRef] - Cowan, M.J.; Pike, K.C.; Budzynski, H.K. Psychosocial Nursing Therapy Following Sudden Cardiac Arrest: Impact on Two-Year Survival. Nurs. Res.
**2001**, 50, 68–76. [Google Scholar] [CrossRef] - Yu, L.C.; Lin, I.M.; Fan, S.Y.; Chien, C.L.; Lin, T.H. One-Year Cardiovascular Prognosis of the Randomized, Controlled, Short-Term Heart Rate Variability Biofeedback Among Patients with Coronary Artery Disease. Int. J. Behav. Med.
**2018**, 25, 271–282. [Google Scholar] [CrossRef] - Zucker, T.L.; Samuelson, K.W.; Muench, F.; Greenberg, M.A.; Gevirtz, R.N. The Effects of Respiratory Sinus Arrhythmia Biofeedback on Heart Rate Variability and Posttraumatic Stress Disorder Symptoms: A Pilot Study. Appl. Psychophysiol. Biofeedback
**2009**, 34, 135–143. [Google Scholar] [CrossRef] - Pizzoli, S.F.M.; Marzorati, C.; Gatti, D.; Monzani, D.; Mazzocco, K.; Pravettoni, G. A Meta-Analysis on Heart Rate Variability Biofeedback and Depressive Symptoms. Sci. Rep.
**2021**, 11, 1–10. [Google Scholar] [CrossRef] - Firth, A.M.; Cavallini, I.; Sütterlin, S.; Lugo, R.G. Mindfulness and Self-Efficacy in Pain Perception, Stress and Academic Performance. The Influence of Mindfulness on Cognitive Processes. Psychol. Res. Behav. Manag.
**2019**, 12, 565–574. [Google Scholar] [CrossRef] [PubMed] - Goessl, V.C.; Curtiss, J.E.; Hofmann, S.G. The Effect of Heart Rate Variability Biofeedback Training on Stress and Anxiety: A Meta-Analysis. Psychol. Med.
**2017**, 47, 2578–2586. [Google Scholar] [CrossRef] - Gevirtz, R. The Promise of Heart Rate Variability Biofeedback: Evidence-Based Applications. Biofeedback
**2013**, 41, 110–120. [Google Scholar] [CrossRef] - Lin, G.; Xiang, Q.; Fu, X.; Wang, S.; Wang, S.; Chen, S.; Shao, L.; Zhao, Y.; Wang, T. Heart Rate Variability Biofeedback Decreases Blood Pressure in Prehypertensive Subjects by Improving Autonomic Function and Baroreflex. J. Altern. Complement. Med.
**2012**, 18, 143–152. [Google Scholar] [CrossRef] [PubMed] - Leganes-Fonteneau, M.; Bates, M.E.; Muzumdar, N.; Pawlak, A.; Islam, S.; Vaschillo, E.; Buckman, J.F. Cardiovascular Mechanisms of Interoceptive Awareness: Effects of Resonance Breathing. Int. J. Psychophysiol.
**2021**, 169, 71–87. [Google Scholar] [CrossRef] - Schwerdtfeger, A.R.; Schwarz, G.; Pfurtscheller, K.; Thayer, J.F.; Jarczok, M.N.; Pfurtscheller, G. Heart Rate Variability (HRV): From Brain Death to Resonance Breathing at 6 Breaths per Minute. Clin. Neurophysiol.
**2020**, 131, 676–693. [Google Scholar] [CrossRef] - Rominger, C.; Graßmann, T.M.; Weber, B.; Schwerdtfeger, A.R. Does Contingent Biofeedback Improve Cardiac Interoception? A Preregistered Replication of Meyerholz, Irzinger, Withoft, Gerlach, and Pohl (2019) Using the Heartbeat Discrimination Task in a Randomised Control Trial. PLoS ONE
**2021**, 16, e0248246. [Google Scholar] [CrossRef] [PubMed] - Bae, D.; Matthews, J.J.L.; Chen, J.J.; Mah, L. Increased Exhalation to Inhalation Ratio during Breathing Enhances High-Frequency Heart Rate Variability in Healthy Adults. Psychophysiology
**2021**, 58, e13905. [Google Scholar] [CrossRef] [PubMed] - Van Diest, I.; Verstappen, K.; Aubert, A.E.; Widjaja, D.; Vansteenwegen, D.; Vlemincx, E. Inhalation/Exhalation Ratio Modulates the Effect of Slow Breathing on Heart Rate Variability and Relaxation. Appl. Psychophysiol. Biofeedback
**2014**, 39, 171–180. [Google Scholar] [CrossRef] - Malik, M.; Bigger, J.T.; Camm, A.J.; Kleiger, R.E.; Malliani, A.; Moss, A.J.; Schwartz, P.J. Heart Rate Variability: Standards of Measurement, Physiological Interpretation, and Clinical Use. Eur. Heart J.
**1996**, 17, 354–381. [Google Scholar] [CrossRef] - Munoz, M.L.; Van Roon, A.; Riese, H.; Thio, C.; Oostenbroek, E.; Westrik, I.; De Geus, E.J.C.; Gansevoort, R.; Lefrandt, J.; Nolte, I.M.; et al. Validity of (Ultra-)Short Recordings for Heart Rate Variability Measurements. PLoS ONE
**2015**, 10, e0138921. [Google Scholar] [CrossRef] [PubMed] - Cohen, S.; Kamarck, T.; Mermelstein, R. A Global Measure of Perceived Stress. J. Health Soc. Behav.
**1983**, 24, 385–396. [Google Scholar] [CrossRef] - Brown, K.W.; Ryan, R.M. The Benefits of Being Present: Mindfulness and Its Role in Psychological Well-Being. J. Pers. Soc. Psychol.
**2003**, 84, 822–848. [Google Scholar] [CrossRef] - Osman, A.; Lamis, D.A.; Bagge, C.L.; Freedenthal, S.; Barnes, S.M. The Mindful Attention Awareness Scale: Further Examination of Dimensionality, Reliability, and Concurrent Validity Estimates. J. Pers. Assess.
**2016**, 98, 189–199. [Google Scholar] [CrossRef] - Mehling, W.E.; Price, C.; Daubenmier, J.J.; Acree, M.; Bartmess, E.; Stewart, A. The Multidimensional Assessment of Interoceptive Awareness (MAIA). PLoS ONE
**2012**, 7, e48230. [Google Scholar] [CrossRef] - Yang, J.; Choudhary, G.I.; Rahardja, S.; Franti, P. Classification of Interbeat Interval Time-Series Using Attention Entropy. IEEE Trans. Affect. Comput.
**2020**, 1–10. [Google Scholar] [CrossRef] - Udhayakumar, R.K.; Karmakar, C.; Palaniswami, M. Approximate Entropy Profile: A Novel Approach to Comprehend Irregularity of Short-Term HRV Signal. Nonlinear Dyn.
**2017**, 88, 823–837. [Google Scholar] [CrossRef] - Udhayakumar, R.K.; Karmakar, C.; Palaniswami, M. Understanding Irregularity Characteristics of Short-Term HRV Signals Using Sample Entropy Profile. IEEE Trans. Biomed. Eng.
**2018**, 65, 2569–2579. [Google Scholar] [CrossRef] [PubMed] - Manis, G.; Sassi, R. A Python Library with Fast Algorithms for Popular Entropy Definitions. In Proceedings of the Computing in Cardiology Conference, Brno, Czech Republic, 13–15 September 2021; Volume 48, pp. 1–4. [Google Scholar] [CrossRef]
- Shen, E.; Cai, Z.; Gu, F. Mathematical Foundation of a New Complexity Measure. Appl. Math. Mech.
**2005**, 26, 1188–1196. [Google Scholar] - Girault, J.M.; Humeau-Heurtier, A. Centered and Averaged Fuzzy Entropy to Improve Fuzzy Entropy Precision. Entropy
**2018**, 20, 287. [Google Scholar] [CrossRef] - Costa, M.; Ghiran, I.; Peng, C.K.; Nicholson-Weller, A.; Goldberger, A.L. Complex Dynamics of Human Red Blood Cell Flickering: Alterations with in Vivo Aging. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2008**, 78, 020901. [Google Scholar] [CrossRef] - Batista, G.E.A.P.A.; Keogh, E.J.; Tataw, O.M.; De Souza, V.M.A. CID: An Efficient Complexity-Invariant Distance for Time Series. Data Min. Knowl. Discov.
**2014**, 28, 634–669. [Google Scholar] [CrossRef] - Wu, S.D.; Wu, C.W.; Lin, S.G.; Wang, C.C.; Lee, K.Y. Time series analysis using composite multiscale entropy. Entropy
**2013**, 15, 1069. [Google Scholar] [CrossRef] - Lake, D.E.; Moorman, J.R. Accurate Estimation of Entropy in Very Short Physiological Time Series: The Problem of Atrial Fibrillation Detection in Implanted Ventricular Devices. Am. J. Physiol. Hear. Circ. Physiol.
**2011**, 300, 319–325. [Google Scholar] [CrossRef] - Chanwimalueang, T.; Mandic, D.P. Cosine Similarity Entropy: Self-Correlation-Based Complexity Analysis of Dynamical Systems. Entropy
**2017**, 19, 652. [Google Scholar] [CrossRef] - Olofsen, E.; Sleigh, J.W.; Dahan, A. Permutation Entropy of the Electroencephalogram: A Measure of Anaesthetic Drug Effect. Br. J. Anaesth.
**2008**, 101, 810–821. [Google Scholar] [CrossRef] - Kugiumtzis, D.; Tsimpiris, A. Measures of Analysis of Time Series (MATS): A MATLAB Toolkit for Computation of Multiple Measures on Time Series Data Bases. J. Stat. Softw.
**2010**, 33, 1–30. [Google Scholar] [CrossRef] - Shi, L.C.; Jiao, Y.Y.; Lu, B.L. Differential Entropy Feature for EEG-Based Vigilance Estimation. In Proceedings of the 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS, Osaka, Japan, 3–7 July 2013; pp. 6627–6630. [Google Scholar] [CrossRef]
- Giannakopoulos, T.; Pikrakis, A. Introduction to Audio Analysis: A MATLAB Approach; Academic Press: Cambridge, MA, USA, 2014. [Google Scholar] [CrossRef]
- Huo, Z.; Zhang, Y.; Shu, L.; Liao, X. Edge Permutation Entropy: An Improved Entropy Measure for Time-Series Analysis. In Proceedings of the 45th IECON Proceedings (Industrial Electronics Conference), Lisbon, Portugal, 14–17 October 2019; Volume 2019, pp. 5998–6003. [Google Scholar] [CrossRef]
- Fernández, N.; Maldonado, C.; Gershenson, C. Information Measures of Complexity, Emergence, Self-Organization, Homeostasis, and Autopoiesis. In Guided Self-Organization: Inception; Springer: Berlin/Heidelberg, Germany, 2014; pp. 19–51. [Google Scholar] [CrossRef]
- Scargle, J.D. Studies in Astronomical Time Series Analysis. II—Statistical Aspects of Spectral Analysis of Unevenly Spaced Data. Astrophys. J.
**1982**, 263, 835. [Google Scholar] [CrossRef] - Cooley, J.W.; Tukey, J.W. An Algorithm for the Machine Calculation of Complex Fourier Series. Math. Comput.
**1965**, 19, 297–301. [Google Scholar] [CrossRef] - Chang, Y.; Peng, L.; Liu, C.; Wang, X.; Yin, C.; Yao, L. Novel Gridded Descriptors of Poincaré Plot for Analyzing Heartbeat Interval Time-Series. Comput. Biol. Med.
**2019**, 109, 280–289. [Google Scholar] [CrossRef] - Liu, X.; Jiang, A.; Xu, N.; Xue, J. Increment Entropy as a Measure of Complexity for Time Series. Entropy
**2016**, 18, 22. [Google Scholar] [CrossRef] - Teixeira, J.P.; Oliveira, C.; Lopes, C. Vocal Acoustic Analysis—Jitter, Shimmer and HNR Parameters. Procedia Technol.
**2013**, 9, 1112–1122. [Google Scholar] [CrossRef] - Zozor, S.; Mateos, D.; Lamberti, P.W. Mixing Bandt-Pompe and Lempel-Ziv Approaches: Another Way to Analyze the Complexity of Continuous-State Sequences. Eur. Phys. J. B
**2014**, 87, 1–12. [Google Scholar] [CrossRef] - Burg, J.P. Maximum Entropy Spectral Analysis. Ph.D. Thesis, Stanford University, Stanford, CA, USA, 1975. [Google Scholar]
- Castiglioni, P.; Faini, A. A Fast DFA Algorithm for Multifractal Multiscale Analysis of Physiological Time Series. Front. Physiol.
**2019**, 10, 115. [Google Scholar] [CrossRef] - Wu, S.D.; Wu, C.W.; Lee, K.Y.; Lin, S.G. Modified Multiscale Entropy for Short-Term Time Series Analysis. Phys. A Stat. Mech. Appl.
**2013**, 392, 5865–5873. [Google Scholar] [CrossRef] - Zunino, L.; Soriano, M.C.; Rosso, O.A. Distinguishing Chaotic and Stochastic Dynamics from Time Series by Using a Multiscale Symbolic Approach. Phys. Rev. E-Stat. Nonlinear Soft Matter Phys.
**2012**, 86, 046210. [Google Scholar] [CrossRef] - Bai, Y.; Liang, Z.; Li, X.; Voss, L.J.; Sleigh, J.W. Permutation Lempel-Ziv Complexity Measure of Electroencephalogram in GABAergic Anaesthetics. Physiol. Meas.
**2015**, 36, 2483–2501. [Google Scholar] [CrossRef] - Lake, D.E. Improved Entropy Rate Estimation in Physiological Data. In Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS, Boston, MA, USA, 30 August–3 September 2011; pp. 1463–1466. [Google Scholar] [CrossRef]
- Little, M.; McSharry, P.; Roberts, S.; Costello, D.; Moroz, I. Exploiting Nonlinear Recurrence and Fractal Scaling Properties for Voice Disorder Detection. Nat. Preced.
**2007**, 1. [Google Scholar] [CrossRef] - Jauregui, M.; Zunino, L.; Lenzi, E.K.; Mendes, R.S.; Ribeiro, H.V. Characterization of Time Series via Rényi Complexity–Entropy Curves. Phys. A Stat. Mech. Appl.
**2018**, 498, 74–85. [Google Scholar] [CrossRef] - Lad, F.; Sanfilippo, G.; Agrò, G. Extropy: Complementary Dual of Entropy. Stat. Sci.
**2015**, 30, 40–58. [Google Scholar] [CrossRef] - Inouye, T.; Shinosaki, K.; Sakamoto, H.; Toi, S.; Ukai, S.; Iyama, A.; Katsuda, Y.; Hirano, M. Quantification of EEG Irregularity by Use of the Entropy of the Power Spectrum. Electroencephalogr. Clin. Neurophysiol.
**1991**, 79, 204–210. [Google Scholar] [CrossRef] - Girault, J.M. Recurrence and Symmetry of Time Series: Application to Transition Detection. Chaos Solitons Fractals
**2015**, 77, 11–28. [Google Scholar] [CrossRef] - Moulder, R.G.; Daniel, K.E.; Teachman, B.A.; Boker, S.M. Tangle: A Metric for Quantifying Complexity and Erratic Behavior in Short Time Series. Psychol. Methods
**2022**, 27, 82–98. [Google Scholar] [CrossRef] - Zunino, L.; Pérez, D.G.; Martín, M.T.; Garavaglia, M.; Plastino, A.; Rosso, O.A. Fractional Brownian Motion, Fractional Gaussian Noise, and Tsallis Permutation Entropy. Phys. A Stat. Mech. Appl.
**2008**, 387, 6057–6068. [Google Scholar] [CrossRef] - Bernaola-Galván, P.A.; Gómez-Extremera, M.; Romance, A.R.; Carpena, P. Correlations in Magnitude Series to Assess Nonlinearities: Application to Multifractal Models and Heartbeat Fluctuations. Phys. Rev. E
**2017**, 96, 032218. [Google Scholar] [CrossRef] [PubMed] - Rosso, O.A.; Blanco, S.; Yordanova, J.; Kolev, V.; Figliola, A.; Schürmann, M.; Baar, E. Wavelet Entropy: A New Tool for Analysis of Short Duration Brain Electrical Signals. J. Neurosci. Methods
**2001**, 105, 65–75. [Google Scholar] [CrossRef] [PubMed] - Kraemer, K.H.; Datseris, G.; Kurths, J.; Kiss, I.Z.; Ocampo-Espindola, J.L.; Marwan, N. A Unified and Automated Approach to Attractor Reconstruction. New J. Phys.
**2021**, 23, 033017. [Google Scholar] [CrossRef] - Grassberger, P. Grassberger-Procaccia Algorithm. Scholarpedia
**2007**, 2, 3043. [Google Scholar] [CrossRef] - Bandt, C.; Pompe, B. Permutation Entropy: A Natural Complexity Measure for Time Series. Phys. Rev. Lett.
**2002**, 88, 174102. [Google Scholar] [CrossRef] [PubMed] - Panday, D.; Mayor, D.; Kandel, H.; ProcessSignals. Detection of Real EGG and BVP Peaks from Noisy Biosignals: An Innovative MATLAB-Based Graphical User Interface (GUI). Abstract for a Poster Presentation, GUI. Available online: http://electroacupuncture.qeeg.co.uk/processsignals (accessed on 28 September 2022).
- McConnell, M.; Schwerin, B.; So, S.; Richards, B. RR-APET—Heart Rate Variability Analysis Software. Comput. Methods Programs Biomed.
**2020**, 185, 105127. [Google Scholar] [CrossRef] - RStudio Team. RStudio: Integrated Development Environment for R; RStudio Team: Boston, MA, USA, 2020. [Google Scholar]
- Pohlert, T. PMCMRplus: Calculate Pairwise Multiple Comparisons of Mean Rank Sums Extended. 2021. Available online: https://CRAN.R-project.org/package=PMCMRplus (accessed on 29 January 2023).
- Patil, I. Visualizations with Statistical Details: The “ggstatsplot” Approach. J. Open Source Softw.
**2021**, 6, 3167. [Google Scholar] [CrossRef] - Datseris, G.; Parlitz, U. Nonlinear Dynamics: A Concise Introduction Interlaced with Code; Springer Nature: Berlin/Heidelberg, Germany, 2022. [Google Scholar] [CrossRef]
- Martinez-Garcia, M.; Zhang, Y.; Wang, S. Enhancing Stochastic Resonance by Adaptive Colored Noise and Particle Swarm Optimization: An Application to Steering Control. In Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, AIM, Sapporo, Japan, 11–15 July 2022; Volume 2022, pp. 1700–1705. [Google Scholar] [CrossRef]
- Beinecke, J.; Heider, D. Gaussian Noise Up-Sampling Is Better Suited than SMOTE and ADASYN for Clinical Decision Making. BioData Min.
**2021**, 14, 1–11. [Google Scholar] [CrossRef] - MATLAB. Interpn: Interpolation for 2-D Gridded Data in Meshgrid Format. Available online: https://uk.mathworks.com/help/matlab/ref/interpn.html (accessed on 8 November 2022).
- Marshall, S. Strip the Willow; Penguin: London, UK, 1997. [Google Scholar]
- MATLAB. Signal Processing Toolbox. Available online: https://uk.mathworks.com/help/signal/index.html?s_tid=hc_product_card (accessed on 20 January 2023).
- Li, P.; Karmakar, C.; Yan, C.; Palaniswami, M.; Liu, C. Classification of 5-S Epileptic EEG Recordings Using Distribution Entropy and Sample Entropy. Front. Physiol.
**2016**, 7, 136. [Google Scholar] [CrossRef] [PubMed] - Marshall, E.; Marquier, B. Friedman Test in SPSS (Non-Parametric Equivalent to Repeated Measures ANOVA). Available online: https://www.sheffield.ac.uk/media/35112/download?attachment (accessed on 20 January 2023).
- Tomczak, M.; Tomczak, E. The Need to Report Effect Size Estimates Revisited. An Overview of Some Recommended Measures of Effect Size. Trends Sport Sci.
**2014**, 1, 19–25. [Google Scholar] - Spencer, N.; (University of Hertfordshire, Hatfield, UK). Personal Communication, 2022.
- Conover, W.J. Practical Nonparametric Statistics, 3rd ed.; Wiley: Hoboken, NJ, USA, 1999. [Google Scholar]
- Laerd, A.; Laerd, M. Linear Regression Analysis in SPSS Statistics—Procedure, Assumptions and Reporting the Output. Available online: https://statistics.laerd.com/spss-tutorials/linear-regression-using-spss-statistics.php (accessed on 22 October 2022).
- Tarvainen, M.P.; Niskanen, J.P.; Lipponen, J.A.; Ranta-aho, P.O.; Karjalainen, P.A. Kubios HRV—Heart Rate Variability Analysis Software. Comput. Methods Programs Biomed.
**2014**, 113, 210–220. [Google Scholar] [CrossRef] [PubMed] - Reyes del Paso, G.A.; Langewitz, W.; Mulder, L.J.M.; van Roon, A.; Duschek, S. The Utility of Low Frequency Heart Rate Variability as an Index of Sympathetic Cardiac Tone: A Review with Emphasis on a Reanalysis of Previous Studies. Psychophysiology
**2013**, 50, 477–487. [Google Scholar] [CrossRef] - Kubios. HRV in Evaluating ANS Function. Available online: https://www.kubios.com/hrv-ans-function/ (accessed on 10 November 2022).
- Baevsky, R.M.; Berseneva, A.P. Anwendungen des System Kardivar zur Feststellung des Stressniveaus und des Anpassungsvermögens des Organismus; Messungsstandards und Physiologische Interpretation: Moscow, Russia; Prague, Czech Republic, 2008. [Google Scholar]
- Li, P.; Liu, C.; Li, K.; Zheng, D.; Liu, C.; Hou, Y. Assessing the Complexity of Short-Term Heartbeat Interval Series by Distribution Entropy. Med. Biol. Eng. Comput.
**2015**, 53, 77–87. [Google Scholar] [CrossRef] - Datseris, G.; Isensee, J.; Pech, S.; Gál, T. DrWatson: The Perfect Sidekick for Your Scientific Inquiries. J. Open Source Softw.
**2020**, 5, 2673. [Google Scholar] [CrossRef] - Sabeti, M.; Katebi, S.; Boostani, R. Entropy and Complexity Measures for EEG Signal Classification of Schizophrenic and Control Participants. Artif. Intell. Med.
**2009**, 47, 263–274. [Google Scholar] [CrossRef] [PubMed] - Grenier, P.; Parent, A.C.; Huard, D.; Anctil, F.; Chaumont, D. An Assessment of Six Dissimilarity Metrics for Climate Analogs. J. Appl. Meteorol. Climatol.
**2013**, 52, 733–752. [Google Scholar] [CrossRef]

**Figure 2.**Example of the participants’ pacer display for a slow-paced breathing trial. The blue line on the left and the bar graph on the right rise and fall at a rate of 6 BrPM, i.e., 10 s per cycle with an inhalation/exhalation ratio of 40/60, or 4 s in and 6 s out, with no pause in breathing.

**Figure 3.**Respiration cycles at a 40/60 inhalation/exhalation ratio, showing INbreath (Blue to Red dot), OUTbreath (Red to Blue dot) and peak-to-peak (PP) (Red to Red dot) respiration intervals.

**Figure 4.**(

**a**) RRi data: Numbers of Baseline to 5 BrPM (Ba_50) and Baseline to RBR (Ba_RBR) trial pairs with > 3 values of the Conover S statistic above the 95th percentile for all pairs, for all durations of data (1 to 5 min), for non-resampled (noR) and resampled (4R) data. The threshold of ‘> 3 values’ of S was selected because, for most comparisons, counts were very low (0 or 1), so that their upper quartile (75th percentile) was 4; (

**b**) numbers of the five trial pairs with most values of the Conover S statistic above the 95th percentile for all pairs, for 5-min data only (no Baseline to 7 BrPM trial pairs met this criterion, nor did any Self-paced to externally paced trial pairs).

**Figure 5.**Examples of ‘Top’ CEPS RRi measures that decrease as breathing frequency increases, by family, with values of Friedman’s χ

^{2}: (

**a**) EPP SD1_4 to SD1_7 (HRA); (

**b**) LLE32-36 (OC); (

**c**) HRV frequency domain measures, from Kubios HRV. Note that values have been standardised to the range (1, 2), for ease of comparison.

**Figure 6.**Examples of ‘Top’ CEPS RRi measures that increase as breathing frequency increases, by family, with values of Friedman’s χ

^{2}: (

**a**) 11 FD measures (FD); (

**b**) mPM_E and CPEI (PE-based); (

**c**) Some other entropies (OE). Note that values have been standardised to the range (1,2), for ease of comparison, and that PJSC, a complexity rather than an entropy measure, decreases as breathing frequency increases.

**Figure 7.**‘Top’ CEPS RSP measures, by family, with values of Friedman’s χ

^{2}. Note that values have been standardised to the range (1,2), for ease of comparison.

**Figure 8.**DynamicalSystems.jl FD measures, with Wavelet entropy (wavent) as a comparator: (

**a**) 4R RRi data; (

**b**) raw RSP data; (

**c**) EDA data. Note that values have been standardised to the range (1,2), for ease of comparison.

**Figure 9.**The effects of paced breathing on: (

**a**) the ‘classical’ heart rate asymmetry (HRA) indices and the Asymmetric Spread Index (ASI); (

**b**) Guzik’s subsidiary descriptors; (

**c**) normalised HRA measures. Ehlers’ Index (Friedman’s χ

^{2}= 92.926) is not shown.

**Figure 10.**Left: Correlations (Spearman’s rho) between measures for 1-min, 2-min, 3-min or 4-min data with the same measures for 5-min data: (

**a**) RRi noR data; (

**b**) RRi (4R) data; (

**c**) RRi (10R) data. Median and IQR of the ICCs for each of the three sets of measures are shown below each Figure part.

**Figure 11.**Percentage differences between median values of the measures for 1-min, 2-min, 3-min or 4-min data and the same measures for 5-min data. Top row (

**a**,

**b**): Two FD measures, for RRi (noR) and RRi (4R) data; Middle row (

**c**,

**d**): Average entropy (AE) and FD_C, for RRi (noR) and RRi (4R) data; Bottom row (

**e**,

**f**): Permutation-based CPEI and FD_PRI for non-resampled data, with RR-APET measures LF power and SDNN. Note that y-axes are all at different scales.

**Figure 12.**Scatter plots of standardised values of ICC and Conover S, for measures based on: (

**a**) RRi (noR) measures; (

**b**) RRi (4R) measures; (

**c**) EDA measures, and (

**d**) HRV measures from RR-APET. In parts (

**a**–

**c**), data points are in red for FDs, in blue for HRAs (in blue) and in green for PE-based measures. In part (

**d**), points in red are for time-domain measures, points in blue for frequency-domain measures, in green for nonlinear complexity measures, and in yellow for the nonlinear RQA measures.

**Figure 13.**Changes in Mann-Whitney effect sizes (RBR vs. ‘rest’) with data length (1- to 5-min): (

**a**,

**b**) fractal dimensions; (

**c**,

**d**) PE-based measures; (

**e**,

**f**) some HRA indices; (

**g**) RR-APET HRV measures (the Welch periodogram method was used for the frequency domain measures). Note that y-axes are not all to the same scale.

**Table 1.**A literature review of fractal dimension measures. Columns show Name, Abbreviation used for each measure, Selected references, numbers of studies located using PubMed and Google Scholar, and date of the first publication located for each measure that included the terms “fractal dimension” AND [Name]. These dates may not, however, indicate first publication of a particular FD measure. Numbers of hits for “[Name OR Name’s] fractal dimension” are shown in parentheses. All the measures listed, except those from Witold Kinsner, have been used in this paper. In this and the following Tables, alternating rows have been given a coloured background simply to aid readability.

Name | Abbrev. | Selected References | PubMed | Google Scholar | ||
---|---|---|---|---|---|---|

N | Date 1st | N | Date 1st | |||

Higuchi | FD_H | Higuchi 1988 [14] | 153 (116) | 1994 | 5180 (1610) | 1988 |

Katz | FD_K | Katz 1988 [15] | 34 (16) | 1994 | 6620 (436) | 1985 |

Castiglioni | FD_C | Castiglioni 2010 [16] | 3 (5) | 2010 | 561 (13) | 2010 |

Mandelbrot | FD_M | Castiglioni 2010 [16] | 33 (42) | 1975 | 46,400 (108) | 1967 |

Petrosian | FD_P | Petrosian 1995 [17] | 5 (8) | 2010 | 876 (296) | 1995 |

Sevcik | FD_S | Sevcik 1998 [18] | 4 (4) | 2009 | 534 (79) | 1998 |

Box-count [Moisy] | FD_Box_M | Moisy 2022 [19] | 370 (40) | 1990 | 34,410 (2029) | c. 1985 |

Meerwijk/ van der Linden | FD_Box_MvdL | Meerwijk et al. 2015 [20] | 2 (0) | 2014 | 11 (0) | 2015 |

Kalauzi | NLDwL NLDwP NLDiL NLDiP | Kalauzi et al. 2009 [8] | 7 (0) | 2005 | 243 (0) | 2009 |

Tamulevičius, Kizlaitienė | FD_Amp FD_Dist FD_Sign FD_LRI FD_PRI | Kizlaitienė 2021 [21] | 0 (0) | n/a | 1 (0) | 2021 |

Maragos | mFD_M | Maragos 1994; [22] Zlatintsi and Maragos 2013 [23] | 2 (0) | 1999 | 829 (4) | 1993 |

Kinsner | D_{β}D _{σ} | Kinsner 2008 [24] | 1 (1) | 2001 | 691 (2) | 1989 |

**Table 2.**A literature review of heart rate asymmetry measures. Columns show Name, Abbreviation used for each measure, Selected references, numbers of studies located using PubMed and Google Scholar, and date of the first publication located for each measure that included the terms “heart rate asymmetry” AND [Name]. These dates may not indicate the first publication of a particular measure. Numbers of hits for “Name OR Name’s Index” are shown in parentheses. All the measures listed have been used in this paper.

Name | Abbrev. | Selected References | PubMed | Google Scholar | ||
---|---|---|---|---|---|---|

N | Date 1st | N | Date 1st | |||

Ehlers’ Index | EI | Ehlers et al. 1998 [35] | 4 (4) | 2009 | 59 (37) | 2006 |

Guzik’s Index | GI | Guzik et al. 2006 [36] | 24 (9) | 2006 | 1 (63) | 2008 |

Porta’s Index | PI | Porta et al. 2006 [38] | 15 (11) | 2012 | 188 (123 ^{a}) | 2006 |

Slope Index (Karmakar) | SI | Karmakar et al. 2012 [39] | 4 (4) | 2015 | 28 (^{b}) | 2012 |

Area Index (Karmakar) | AI | Yan et al. 2017 [40] | 3 (2 ^{b}) | 2017 | 17 (^{b}) | 2017 |

Asymmetric Spread Index (Rohila) | ASI | Rohila and Sharma 2020 [42] | 0 | n/a | 2 | 2020 |

Deceleration contributions | SD1_{up}, SD2_{up} | Guzik et al. 2006 2006 [36] | 1 | 2007 | 8, 0 (69, 12 ^{b}) | 2006 |

Acceleration contributions | SD1_{down},SD2 _{down} ^{c} | Guzik et al. 2006 2006 [36] | 0 | n/a | 0, 0 (23, 12 ^{b}) | n/a |

SD1_{up}^{2}/SD1^{2},SD2 _{up}^{2}/SD2^{2} | C1_{a}, C2_{a} | Guzik et al. 2006 2006 [36] (adapted by Rohila) ^{d} | 1 | 2022 | 5 | 2021 |

SD1_{dn}^{2}/SD1^{2},SD2 _{dn}^{2}/SD2^{2} | C1_{d}, C2_{d} | Guzik et al. 2006 2006 [36] (adapted by Rohila) | 1 | 2022 | 5 | 2021 |

√((SD1_{up}^{2} + SD2_{up}^{2})/2) | SDNN_{up} | Piskorski and Guzik 2012 [48] | 1 | 2022 | 17 | 2021 |

√((SD1_{down}^{2} + SD2_{down}^{2})/2) | SDNN_{down} | Piskorski and Guzik 2012 [48] | 1 | 2022 | 17 | 2021 |

^{a}. There were 164 hits for ‘Porta Index”, and 41 for “La Porta Index”;

^{b}. Terms such as “Slope Index” and “Area Index” are used in many contexts, so that searching for them without some form of qualifier such as author Name was not helpful;

^{c}. Some authors now use SD1

_{low}and SD2

_{low}instead of SD1

_{down}and SD2

_{down}[49]; d. Guzik et al. originally used C

_{up}and C

_{dn}for SD1

_{up}

^{2}/SD1

^{2}and SD1

_{dn}

^{2}/SD1

^{2}. The corresponding indices for SD2 were added in a later paper, along with the change to C

_{a}and C

_{d}[48]. We have used SDNN

_{up}and SDNN

_{down}rather than SDNN

_{a}and SDNN

_{d}. In this and the other Tables in this paper, alternating rows have been given a coloured background simply to aid readability.

Age | Female | Male | All |
---|---|---|---|

18–24 | 2 | 4 | 6 |

25–34 | 3 | 6 | 9 |

35–44 | 6 | 4 | 10 |

45–54 | 3 | 5 | 8 |

55–64 | 3 | 3 | 6 |

65–74 | 0 | 3 | 3 |

75–84 | 1 | 1 | 2 |

Total | 18 | 26 | 44 |

**Table 4.**Other measures newly implemented in CEPS 2, or in course of implementation (asterisked). Measures planned for future inclusion are listed in parentheses (for measures already included in CEPS, see [4]). Measures are listed in alphabetical order, showing original references, names of code providers, code type and institutions of originators. Please note that, although every effort has been made to implement these measures correctly in CEPS, time has not always allowed us to validate the results obtained when using CEPS with those researchers who provided us with code. As with all Creative Commons licensing, CEPS is provided freely and without warranty, on condition that this paper is referenced in any outputs that result from using the software.

Measure | Original Author/s | Provider | Source Code | Institution |
---|---|---|---|---|

AttnEn | Yang et al. 2020 [80] | EntropyHub | MATLAB | Xi’an |

AvgApEnP | Udhayakumar et al. 2017 [81] | Karmakar | MATLAB | Melbourne |

AvgSampEnP | Udhayakumar et al. 2018 [82] | Karmakar | MATLAB | Melbourne |

(B_ApEn) | Manis and Sassi 2021 [83] | Published paper | Python | Ioannina/ Milano |

(B_SampEn) | Manis and Sassi 2021 [83] | Published paper | Python | Ioannina/ Milano |

(C0) | Shen et al. 2005 [84] | (Panday) | tbc | Fudan |

CAFE | Girault and Humeau-Heurtier 2018 [85] | Girault | MATLAB | Angers |

CI * | Costa et al. 2008 [86] | Panday | MATLAB | Harvard |

(CID) | Batista et al. 2013 [87] | Published paper | MATLAB | California (Riverside) |

CmSE | Wu et al. 2013 [88] | Published paper | MATLAB | Taipei |

CoSEn | Lake 2011 [89] | Liu | MATLAB | Virginia (Charlottesville) |

CoSiEn | Chanwimalueang and Mandic 2017 [90] | EntropyHub | MATLAB | Imperial (London) |

CPEI | Olofsen et al. 2008 [91] | Published paper | MATLAB | Leiden/ Auckland |

DE * | Grigolini et al. 2001 [10] | Culbreth | MATLAB | North Texas (Denton) |

DFA Alpha | Kugiumtzis and Tsimpiris 2010 [92] | Published paper | MATLAB | Thessaloniki |

DiffEn * | Shi et al. 2013 [93] | (Panday) | MATLAB | Shanghai |

(EE) | Giannakopoulos and Pikrakis [94] | Mathworks | MATLAB | Agia Paraskevi |

EPE | Huo et al. 2019 [95] | Huo | MATLAB | Lincoln |

ESCHA * | Fernández et al. 2014 [96] | Santamaría Bonfil | R | CONACYT-INEEL, Cuernavaca |

FastLomb * | Scargle 1982 [97] | Mathworks | MATLAB | California (Berkeley) |

FFT * | Cooley and Tukey 1965 [98] | Mathworks | MATLAB | IBM, New York |

GPP * | Platiša et al. 2022 [9] | Kalauzi | MATLAB | Belgrade |

GridEn | Yan et al. 2019 [99] | EntropyHub | MATLAB | Shandong |

IncrEn | Liu et al. 2016 [100] | EntropyHub | MATLAB | Changzhou |

Jitter_Jitt | Teixeira et al. 2013 [101] | Teixeira | MATLAB | Bragança |

Jitter_Jitta | Teixeira et al. 2013 [101] | Teixeira | MATLAB | Bragança |

Jitter_ppq5 | Teixeira et al. 2013 [101] | Teixeira | MATLAB | Bragança |

Jitter_RAP | Teixeira et al. 2013 [101] | Teixeira | MATLAB | Bragança |

L_ApEn * | Manis and Sassi 2021 [83] | Published paper | Python | Ioannina/ Milano |

L_SampEn * | Manis and Sassi 2021 [83] | Published paper | Python | Ioannina/ Milano |

LZPC * | Zozor et al. 2014 [102] | GitHub | C | Grenoble/ Córdoba |

MESA * | Burg 1975 [103] | Dowse | MATLAB | Stanford |

mFmDFA * | Castiglioni and Faini 2019 [104] | Castiglioni | MATLAB | Milano |

MmSE | Wu et al. 2013 [105] | Published paper | MATLAB | Taipei |

mPhEn | Panday n.p. | Panday | MATLAB | Hertfordshire |

PJSC | Zunino et al. 2012 [106] | Zunino | MATLAB | La Plata |

PLZC * | Bai et al. 2015 [107] | Published paper | MATLAB | Yanshan |

QSE * | Lake 2011 [108] | (Panday) | MATLAB | Virginia (Charlottesville) |

(RPDE) | Little et al. 2007 [109] | GitHub: hctsa | MATLAB | Oxford |

RPE | Jauregui et al. 2018 [110] | Zunino | MATLAB | Maringá |

SEx | Lad et al. 2015 [111] | Sanfilippo/Panday | MATLAB | Canterbury, NZ/Palermo |

Shimmer_Shim | Teixeira et al. 2013 [101] | Teixeira | MATLAB | Bragança |

Shimmer_ShdB | Teixeira et al. 2013 [101] | Teixeira | MATLAB | Bragança |

Shimmer_apq3 | Teixeira et al. 2013 [101] | Teixeira | MATLAB | Bragança |

Shimmer_apq5 | Teixeira et al. 2013 [101] | Teixeira | MATLAB | Bragança |

SpEn | Inouye et al. 1991 [112] | Mathworks | MATLAB | Osaka |

SQA * | Girault 2015 [113] | Girault | MATLAB | Angers |

SymDyn * | Various (see Primer) | (Panday) | MATLAB | Various |

(Tangle) | Moulder et al. 2022 [114] | GitHub | R | Virginia (Charlottesville) |

TPE | Zunino et al. 2008 [115] | Zunino | MATLAB | La Plata |

VM * | Bernaola- Galván et al. 2017 [116] | Bernaola- Galván/ Panday | Fortran | Málaga |

(WE) | Rosso et al. 2001 [117] | Mathworks | MATLAB | Buenos Aires |

Data Type | 95%/N | FD | HRA | PE | RQA | OC | OE | ALL |
---|---|---|---|---|---|---|---|---|

noR | 95th % | 11.348 | 9.927 | 11.908 | 5.223 | 6.761 | 7.178 | 8.838 |

N | 22 | 40 | 8 | 17 | 51 | 54 | 192 | |

RRi 4R | 95th % | 9.697 | 8.848 | 9.971 | 5.864 | 5.267 | 8.831 | 9.038 |

N | 22 | 40 | 10 | 19 | 51 | 54 | 196 | |

RRi 10R | 95th % | 8.288 | 9.566 | 9.384 | 6.229 | 9.414 | 9.262 | 9.089 |

N | 22 | 40 | 8 | 19 | 48 | 51 | 188 |

**Table 6.**5-min RRi data: median values of Friedman’s χ

^{2}and Kendall’s W for six groups of CEPS measures, with interquartile ranges (IQRs) in parentheses.

Data Type | χ^{2}/W | FD | HRA | PE | RQA | OC | OE | ALL |
---|---|---|---|---|---|---|---|---|

noR | χ^{2} | 96.597 | 106.888 | 139.596 | 28.953 | 55.626 | 50.863 | 62.795 |

W | 0.310 | 0.350 | 0.448 | 0.093 | 0.178 | 0.163 | 0.202 | |

RRi 4R | χ^{2} | 78.813 | 92.505 | 114.482 | 33.286 | 103.706 | 86.055 | 78.813 |

W | 0.253 | 0.297 | 0.367 | 0.103 | 0.333 | 0.276 | 0.253 | |

RRi 10R | χ^{2} | 63.412 | 113.907 | 108.155 | 38.291 | 102.892 | 103.603 | 95.832 |

W | 0.203 | 0.365 | 0.347 | 0.123 | 0.330 | 0.332 | 0.307 |

**Table 7.**Results for the best-performing RRi measures (values of χ

^{2}> 150), with measures from Kubios HRV provided as a comparison in the lower part of the Table.

Data Type | Measures and χ^{2} Range | FD | HRA | PE | RQA | OC | OE | Best |
---|---|---|---|---|---|---|---|---|

noR | Measures | mFD_M FD_PRI NLDw_mL NLDw_mP FD_C FD_Dist | EPP SD1 (4–7) | CPEI mPM_E PJSC | n/a | n/a | T_E_Ent | mFD_M |

χ^{2} range | 161.208–200.023 | 151.920–178.916 | 150.519–161.949 | 173.660 | 200.023 | |||

RRi 4R | Measures | FD_PRI FD_H mFD_M | n/a | n/a | n/a | LLE32–36 | FE MmSE2, MmSE5 AE | FD_PRI |

χ^{2} | 167.676–197.728 | 156.084–170.126 | 150.084–158.552 | 197.728 | ||||

RRi 10R | Measures | FD_PRI | n/a | n/a | n/a | n/a | BE | FD_PRI |

χ^{2} | 198.906 | 158.334 | 198.906 | |||||

Kubios HRV | General | HRA | Time domain | Freq domain | OC | OE | Best | |

RRi 4R | Measures | PLFP | SD2 SD2/SD1 | SDNN | LFpwr (AR/LS) Totpwr (AR/LS) | DFA alpha1 | SampEnApEn | PLFP [AR LFpwr] |

χ^{2} | 201.714 | 164.605–165.441 | 155.624 | 154.964–181.803 | 167.895 | 155.865–157.280 | 201.714 [181.803] |

**Table 8.**CEPS measures for the 5-min RRi data with standardised values of Conover S for the Baseline-RBR pair ≥ 0.8 are shown below, with non-standardised values of Conover S in parentheses. In

**bold**, values of Conover S for those measures with standardised ICC also ≥0.8. RRi 10R measures with numbers of increases or decreases < 35 are not included. ‘↑’ indicates measure increased between baseline and RBR, and ‘↓’ that it decreased, for the number of participants included in parentheses.

5-min RRi | Measures | noR | 4R | 10R |
---|---|---|---|---|

Baseline-RBR ↑ | SDNNdown SD2down PJSC T_E_ENT EPP SD1_3 EPP SD1_4 EPP SD1_5 EPP SD1_6 EPP SD1_7 EPP SD2_5 AE | 0.809 (12.829) (43 ↑)0.879 (13.934) (44 ↑) 0.914 (14.483) (43 ↑)0.868 (13.758) (43 ↑)0.913 (14.470) (43 ↑)0.915 (14.502) (43 ↑)0.956 (15.151) (43 ↑)0.966 (15.310) (43 ↑) | 0.824 (12.733) (43 ↑)0.830 (12.828) (43 ↑)0.801 (12.392) (43 ↑)0.955 (14.084) (42 ↑) | 0.806 (12.624) (43 ↑) 0.806 (12.624) (43 ↑) (6.796) (35 ↑) (7.899) (39 ↑) (11.964) (43 ↑) (9.074) (39 ↑) |

Baseline-RBR ↓ | FD_C FD_H mFD_M FD_PRI EPP R6 CPEI EPE ImPE mPE1 mPM_E TPE FE MmSE2 | 0.893 (15.151) (44 ↓) 1.000 (15.841) (44 ↓)0.904 (14.327) (43 ↓)0.933 (14.790) (44 ↓) 0.847 (13.423) (43 ↓)0.845 (13.387) (44 ↓)0.811 (12.859) (43 ↓)0.811 (12.859) (43 ↓)0.812 (12.877) (43 ↓)0.856 (13.562) (43 ↓)0.809 (12.826) (43 ↓) | 0.851 (13.151) (44 ↓)0.994 (15.356) (44 ↓)1.000 (15.453) (44 ↓)1.000 (15.392) (44 ↓) 0.800 (12.364) (44 ↓) 0.821 (12.693) (43 ↓) | (7.534) (37 ↓) (10.150) (43 ↓) (12.106) (44 ↓) 1.000 (15.541) (44 ↓) (8.641) (40 ↓) (10.304) (41 ↓) (9.903) (42 ↓) (9.903) (42 ↓) (10.301) (43 ↓) (8.820) (40 ↓) (10.59) (42 ↓) (4.380) (35 ↓) (9.609) (41 ↓) |

Data Type | Measures and χ^{2} Range | FD | HRA | PE | RQA | OC | OE | Best |
---|---|---|---|---|---|---|---|---|

INbreath | Measures | 3 | n/a | 7 | n/a | 18 | 18 | MmSE13 |

χ^{2} range | 155.115–166.382 | 154.082–187.089 | 150.490–190.092 | 159.045–192.043 | 192.043 | |||

OUTbreath | Measures | 3 | 7 | 7 | 1 | 17 | 19 | MmSE13 |

χ^{2} | 157.528–159.289 | 151.338–167.358 | 160.749–188.632 | 167.753 | 151.639–188.884 | 150.950–195.610 | 195.610 | |

Peak-Peak (PP) | Measures | 1 | 8 | 7 | n/a | 17 | 22 | ImPE |

χ^{2} | 168.048 | 177.494–184.878 | 171.671–191.663 | 150.569–191.249 | 152.381–190.565 | 191.663 | ||

Raw RSP | Measures | FD_PRI | n/a | n/a | n/a | n/a | n/a | FD_PRI |

χ^{2} | 154.064 | 154.064 |

**Table 10.**Top two Friedman test results for differences in CEPS measures among all eight trials, for the RRi, RSP and EDA data, with corresponding results for the Kubios HRV measures.

Data Type | Measures | Friedman’s χ^{2} | Kendall’s W |
---|---|---|---|

RRi (noR) | mFD_M FD_H | 200.023 195.703 | 0.642 0.628 |

RRi (4R) | FD_PRI LLE34 | 197.728 170.126 | 0.633 0.546 |

RRi (10R) | FD_PRI BE | 198.906 158.334 | 0.637 0.508 |

RSP (IN) | LLE42 EoD | 190.092 188.255 | 0.251 0.600 |

RSP (OUT) | LLE43 TPE | 188.884 188.642 | 0.250 0.605 |

RSP (PP) | ImPE ESCHA_d | 191.663 191.249 | 0.613 0.612 |

RSP (Raw) | FD_PRI FD_LRI | 154.064 122.040 | 0.493 0.389 |

EDA | RMSSD EPP SD1_1 | 29.035 28.824 | 0.093 0.924 |

Kubios HRV | PLFP LFpwr (AR) | 201.714 181.803 | 0.647 0.583 |

**Table 11.**Top two Friedman test results for differences in DynamicalSystems.jl measures among all eight trials, for the RRi (4R), deduplicated Raw RSP and deduplicated detrended EDA data.

Data Type | Measures | Friedman’s χ^{2} | Kendall’s W |
---|---|---|---|

RRi (4R) | Wavent Perment4 | 135.429 118.979 | 0.440 0.386 |

RSP (Raw) | Wavent Perment4 | 66.910 51.835 | 0.217 0.168 |

EDA | Wavent Delta2 | 16.609 15.214 | 0.054 0.049 |

**Table 12.**‘Top five’ non-standardised values of Conover S for the ECG RRi data, for all 28 pairs of trials.

4R 5-min | Pair | S | 10R 5-min | Pair | S | NoR 5-min | All Base_5 |
---|---|---|---|---|---|---|---|

FD_PRI | Base_5 | 19.013 | FD_PRI | Base_5 | 19.508 | mFD_M | 19.334 |

FD_PRI | Base_5.5 | 15.826 | FD_PRI | Base_5.5 | 15.614 | FD_PRI | 19.163 |

mFD_M | Base_RBR | 15.453 | FD_PRI | Base_RBR | 15.541 | FD_H | 18.672 |

FD_PRI | Base_RBR | 15.392 | MmSE11 | Base_5 | 13.933 | NLDwL_m | 16.566 |

FD_H | Base_RBR | 15.356 | MmSE10 | Base_5 | 13.888 | NLDwP_m | 16.521 |

Medians | 15.453 | 15.541 | 18.672 |

**Table 13.**‘Top five’ non-standardised values of Conover S for the breathing interval data, for all 28 pairs of trials.

IN 5-min | All Base_5 | OUT 5-min | All Base_5 | PP 5-min | All Base_5 |
---|---|---|---|---|---|

IncrEn | 18.037 | ImPE | 17.907 | ImPE | 18.302 |

EoD | 17.898 | Discrete_CS | 17.873 | Discrete_CS | 18.224 |

KLD | 17.898 | IncrEn | 17.867 | EoD | 18.096 |

ImPE | 17.849 | TPE | 17.833 | KLD | 18.096 |

Discrete_CS | 17.713 | mPM_E | 17.794 | mPM_E | 18.057 |

Medians | 17.898 | 17.867 | 18.096 |

**Table 14.**‘Top five’ non-standardised values of Conover S for the raw respiration (RSP) and EDA data, for all 28 pairs of trials.

RSP 5-min | Pair | S | EDA 5-min | Pair | S |
---|---|---|---|---|---|

FD_PRI | Base_5 | 14.349 | GridEn | Base_6 | 5.250 |

FD_PRI | Base_RBR | 12.08 | Jitta | Base_5 | 4.829 |

FD_PRI | Base_5.5 | 11.620 | RMSSD | Base_5 | 4.730 |

FD_PRI | 7_5 | 9.627 | EPP SD1_1 | Base_5 | 4.728 |

FD_LRI | Base_5 | 11.091 | EPP SD1_2 | Base_5 | 4.728 |

Medians | 11.620 | 4.730 |

**Table 15.**Median values of Conover S for four pairs of trials, with lowest values in each row indicated by bold type.

Conover S | Self to RBR | Base to RBR | Self to 5 BrPM | Base to 5 BrPM |
---|---|---|---|---|

RRi (4R) (225) | 4.601 | 7.386 | 4.091 | 7.129 |

RRi (10R) (209) | 5.267 | 8.536 | 5.007 | 8.538 |

RRi (noR) (219) | 3.673 | 6.105 | 3.182 | 6.084 |

RSP raw (99) | 2.549 | 3.307 | 3.069 | 3.277 |

RSP_IN (196) | 2.770 | 7.507 | 4.270 | 8.723 |

RSP_OUT (197) | 4.250 | 8.070 | 4.676 | 9.094 |

RSP_PP (197) | 4.180 | 7.794 | 5.389 | 9.548 |

EDA (89) | 0.812 | 1.732 | 0.892 | 1.865 |

**Table 16.**Measures with standardised median values of χ

^{2}, Kendall’s W and the Baseline-RBR Conover statistic S > 0.8, for two RRi data types and RR-APET. Measures with values lower than any one of these thresholds for one or more of the 2-, 3-, 4- or 5-min data segments are not included. Values within each cell are in order (top to bottom) χ

^{2}, W and S. Total numbers of measures analysed for each data type are shown in parentheses.

RRi (noR) (219) | RRi (4R) (224) | RR-APET (25) | |||
---|---|---|---|---|---|

mFD_M | 194.4 0.624 13.327 | SD2down | 185.6 0.595 12.828 | SD2 | 139.7 0.448 12.319 |

FD_H | 187.8 0.602 15.239 | mFD_M | 156.8 0.503 13.562 | SDNN | 136.7 0.439 12.132 |

FD_PRI | 187.7 0.601 14.790 | Alpha1 | 132.8 0.426 10.251 | ||

EPP SD1_7 | 176.2 0.565 14.691 | ||||

NLDwL_m | 172.1 0.552 14.435 | ||||

NLDwP_m | 171.8 0.551 14.541 | ||||

EPP SD1_6 | 170.7 0.548 15.300 | ||||

FD_C | 161.8 0.519 13.508 | ||||

CPEI | 157.0 0.504 13.043 |

**Table 17.**‘Top 13′ measures for each RRi data type, with most differences of the same sign across the 1-, 2-, 3-, 4- and 5-min data, together with the ‘top 4′ HRV indices. Counts are of increases (↑) or decreases (↓) between Baseline and RBR. Counts excluding the 1-min data are shown in parentheses. Measures for which ICC > 0.9 are in bold type, and total numbers of measures analysed for each data type are shown in parentheses.

RRi (noR) (220) | RRi (4R) (220) | RR-APET (24) | |||
---|---|---|---|---|---|

PJSC (↑) | 199 (162) | FD_C (↓) | 191 (157) | SD2 (↑) | 215 (172) |

RoCV (↑) | 191 (154) | mFD_M (↓) | 191 (155) | SDNN (↑) | 214 (172) |

EPP SD1_6 (↑) | 188 (150) | AE (↑) | 191 (154) | Alpha1 (↑) | 213 (171) |

ACR5 (↓) | 187 (151) | RCmDE7 (↓) | 190 (156) | LFpwr (↑) | 211 (170) |

EPP SD1_5 (↑) | 187 (149) ^{a} | Q3 (↑) | 188 (153) | ||

mPE (↓) | 186 (153) | FD_H (↓) | 187 (152) | ||

EPP r5 (↓) | 186 (150) | LLE32 (↑) | 187 (150) | ||

EPE (↓) | 185 (152) | LLE33 (↑) | 187 (149) | ||

ImPE (↓) | 185 (152) | RoCV (↑) | 185 (149) | ||

AE (↑) | 185 (149) ^{a} | LLE31 (↑) | 183 (146) | ||

EoD (↓) | 184 (151) | RCmDE6 (↓) | 182 (151) | ||

KLD (↓) | 184 (151) | SD2 and _{down}SDNN (↑) _{down} | 182 (146) | ||

MPM_E (↓) | 184 (149) | LLE30 (↑) | 182 (145) |

^{a}. EPP SD1_5 and AE were not in the ‘top 13′ of counts excluding the 1-min data.

**Table 18.**Measures with short-data values consistently within ± 5% of their values for the 5-min data, with those measures least often within ± 5% of their values for the 5-min data.

Top 12 Measures | ICC | Median CV | Count |
---|---|---|---|

FD_H (NoR) | 0.947 | 0.003 | 32 |

NLDwL_m (NoR) | 0.944 | 0.001 | 32 |

NLDwP_m (NoR) | 0.944 | 0.001 | 32 |

Q3 (4R) | 0.910 | 0.008 | 32 |

CPEI (NoR) | 0.897 | 0.008 | 32 |

mFD_M (4R) | 0.894 | 0.016 | 32 |

mFD_M (NoR) | 0.889 | 0.003 | 32 |

LLE33 (4R) | 0.767 | 0.011 | 32 |

LLE32 (4R) | 0.747 | 0.011 | 32 |

Alpha1 | 0.823 | 0.018 | 31 |

LLE30 (4R) | 0.737 | 0.012 | 31 |

LLE31 (4R) | 0.730 | 0.014 | 31 |

Bottom five measures | ICC | Median CV | Count |

PJSC (NoR) | 0.786 | 0.038 | 15 |

EoD (NoR) | 0.849 | 0.069 | 11 |

KLD (NoR) | 0.849 | 0.069 | 11 |

ACR5 (NoR) | 0.864 | 0.124 | 2 |

EPP R5 (NoR) | 0.862 | 0.138 | 2 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mayor, D.; Steffert, T.; Datseris, G.; Firth, A.; Panday, D.; Kandel, H.; Banks, D.
Complexity and Entropy in Physiological Signals (CEPS): Resonance Breathing Rate Assessed Using Measures of Fractal Dimension, Heart Rate Asymmetry and Permutation Entropy. *Entropy* **2023**, *25*, 301.
https://doi.org/10.3390/e25020301

**AMA Style**

Mayor D, Steffert T, Datseris G, Firth A, Panday D, Kandel H, Banks D.
Complexity and Entropy in Physiological Signals (CEPS): Resonance Breathing Rate Assessed Using Measures of Fractal Dimension, Heart Rate Asymmetry and Permutation Entropy. *Entropy*. 2023; 25(2):301.
https://doi.org/10.3390/e25020301

**Chicago/Turabian Style**

Mayor, David, Tony Steffert, George Datseris, Andrea Firth, Deepak Panday, Harikala Kandel, and Duncan Banks.
2023. "Complexity and Entropy in Physiological Signals (CEPS): Resonance Breathing Rate Assessed Using Measures of Fractal Dimension, Heart Rate Asymmetry and Permutation Entropy" *Entropy* 25, no. 2: 301.
https://doi.org/10.3390/e25020301