# The Structure of Chaos: An Empirical Comparison of Fractal Physiology Complexity Indices Using NeuroKit2

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## Abstract

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## 1. Introduction

## 2. Methods

## 3. Results

#### 3.1. Computation Time

#### 3.2. Correlation

#### 3.3. Factor Analysis

#### 3.4. Hierarchical Clustering and Connectivity Network

#### 3.5. Indices Selection

- CWPEn: The Conditional Weighted Permutation Entropy is based on the difference of weighted entropy between that obtained at an embedding dimension m and that obtained at $m+1$ [18].
- LL: The Line Length index stems out of a simplification of Katz’ fractal dimension (KFD) algorithm [19] and corresponds to the average of consecutive absolute differences. It is equivalent to NDLFD, the Fractal dimension via Normalized Length Density [20]. As it captures the amplitude 1-lag fluctuations, this index is likely sensitive to noise in the series.
- BubbEn: The Bubble Entropy is based on Permutation Entropy. It uses the Bubble sort algorithm and counts the number of swaps each vector undergoes in the embedding space instead of ranking their order [21].
- MSWPEn: The Multiscale Weighted Permutation Entropy is the entropy of weighted ordinal descriptors of the time-embedded signal computed at different scales obtained by a coarse-graining procedure [22].
- MFDFA (Max): The value of singularity spectrum D corresponding to the maximum value of singularity exponent H.
- Hjorth: Hjorth’s Complexity is defined as the ratio of the mean frequency of the first derivative of the signal to the mean frequency of the signal [23].
- SVDEn: The Singular Value Decomposition (SVD) Entropy quantifies the amount of eigenvectors needed for an adequate representation of the system [24].
- MFDFA (Width): The width of the multifractal singularity spectrum [25] obtained via Detrended Fluctuation Analysis (DFA).
- MFDFA (Mean): The mean of the maximum and minimum values of singularity exponent H, which quantifies the average fluctuations of the signal.
- MFDFA (Peak): The value of the singularity exponent H corresponding to peak of singularity dimension D. It is a measure of the self-affinity of the signal, and a high value is an indicator of high degree of correlation between the data points.
- MFDFA (Increment): The cumulative function of the squared increments of the generalized Hurst’s exponents between consecutive moment orders [26].
- AttEn: The Attention Entropy is based on the frequency distribution of the intervals between the local maxima and minima of the time series [27].

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Different types of simulated signals, to which was added 5 types of noise (violet, blue, white, pink, and brown) with different intensities. For each signal type, the top row shows the signal with a minimal amount of noise, and the bottom row with a maximal amount of noise.

**Figure 2.**Median computation time difference between the different complexity indices algorithms, as well as variability as a function of signal lengths (represented by different line colors). The indices are grouped in sections (background color) according to their median computation time. Note that the time is expressed in arbitrary units as it is intended to convey differences, since the actual time would depend on the system specifications.

**Figure 4.**Factor loadings of the complexity indices, colored by the factor they represent the most (center). On the left, the median computation times and on the right, the archetypicity—the inverse of factor profile complexity (i.e., the extent to which each index is a pure representative of its dominant factor, which is low for indices that equally load on different factors).

**Figure 5.**Correlation network of the complexity indices. Only the links where |r| > 0.6 are displayed.

**Figure 7.**Variance of the whole dataset of indices explained by the subselection. Each line represents a random number of selected variables. The green line represents the optimal order (i.e., the relative importance) that maximizes the variance explained. The dotted blue line represents the cumulative relative median computation time of the selected indices, and shows that MFDFA and multiscale indices are the most resource-costly algorithms.

**Figure 8.**Visualization of the expected value of a selection of indices depending on the signal type and of the amount of noise.

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**MDPI and ACS Style**

Makowski, D.; Te, A.S.; Pham, T.; Lau, Z.J.; Chen, S.H.A.
The Structure of Chaos: An Empirical Comparison of Fractal Physiology Complexity Indices Using NeuroKit2. *Entropy* **2022**, *24*, 1036.
https://doi.org/10.3390/e24081036

**AMA Style**

Makowski D, Te AS, Pham T, Lau ZJ, Chen SHA.
The Structure of Chaos: An Empirical Comparison of Fractal Physiology Complexity Indices Using NeuroKit2. *Entropy*. 2022; 24(8):1036.
https://doi.org/10.3390/e24081036

**Chicago/Turabian Style**

Makowski, Dominique, An Shu Te, Tam Pham, Zen Juen Lau, and S. H. Annabel Chen.
2022. "The Structure of Chaos: An Empirical Comparison of Fractal Physiology Complexity Indices Using NeuroKit2" *Entropy* 24, no. 8: 1036.
https://doi.org/10.3390/e24081036