# Entropy in Natural Time and the Associated Complexity Measures

## Abstract

**:**

## 1. Introduction

## 2. Natural Time and Natural Time Entropy

#### 2.1. Analysis of Complex Time Series in Natural Time

#### 2.2. Entropy in Natural Time

#### 2.3. Complexity Measures Based on the Entropy in Natural Time

#### 2.3.1. Complexity Measures Based on S

#### 2.3.2. Complexity Measures Based on $\mathrm{\Delta}S$

## 3. Applications of the Natural Time Entropy in Various Complex Systems

#### 3.1. Results for the Electric and Magnetic Signals that Precede Rupture

#### 3.2. Results for the Penetration of Magnetic Flux Avalanches in Type II Superconductors

#### 3.3. Olami–Feder–Christensen Earthquake Model

#### 3.4. Earthquakes

#### 3.5. Electrocardiograms

- When analyzing the RR and QRS intervals of the ECG in natural time, the subjects suffering from sudden cardiac death (SCD) violate [22] one or more of the four healthy limits related to ${\nu}_{s}\left(RR\right)$, ${\nu}_{L}\left(RR\right)$, ${\nu}_{s}\left(QRS\right)$ and ${\nu}_{L}\left(QRS\right)$.
- When analyzing the QT intervals of the ECG in natural time $\delta {S}_{l}\left(QT\right)$ for $l=3$–8 heartbeats, the SCD subjects exhibit [26] almost one order of magnitude larger fluctuations than those of the healthy ones.
- When employing $\mathrm{\Delta}S$ in the analysis of the RR and NN intervals of long duration ECG of SCD patients, the fluctuations of $\mathrm{\Delta}{S}_{7}$ appear to maximize [27] during the last three hours before the ventricular fibrillation.
- When employing $\mathrm{\Delta}S$ in the analysis of the RR and NN intervals of long-duration ECG, N${}_{3}\left(\mathrm{RR}\right)$, N${}_{3}\left(\mathrm{NN}\right)$ together with $\mathrm{\Delta}{S}_{7}$(RR) and $\mathrm{\Delta}{S}_{7}$(NN) may allow the separation [27] of individuals into four classes: healthy, SCD, congestive heart failure (CHF) and atrial fibrillation (AF) individuals.
- The analysis suggested in the previous point is also valid [28] for models of healthy or patient heart dynamics.
- When analyzing the NN intervals of long-duration ECG recordings, the combination of $\sigma \left[\mathrm{\Delta}{S}_{7}\right]$ with ${\mathrm{\Lambda}}_{7}$ and ${\mathrm{\Lambda}}_{49}$ enables (see Table 1 of [29]) the ternary distinction in healthy, SCD and CHF individuals.

#### 3.6. Atmospheric Physics

#### 3.6.1. Ozone Hole Dynamics over Antarctica

#### 3.6.2. Forecasting the Intensity of El Niño/La Niña Southern Oscillation

## 4. Discussion and Perspectives

## 5. Materials and Methods

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

3D | 3-dimensional |

AF | Atrial fibrillation |

CHF | Congestive heart failure |

ECG | Electrocardiograms |

EM | Electromagnetic |

ENSO | El Niño/La Niña Southern Oscillation |

EQ | Earthquake |

GR | Gutenberg–Richter |

OFC | Olami–Feder–Christensen |

NTA | Natural time analysis |

Probability density function | |

SCD | Sudden cardiac death |

SES | Seismic electric signals |

SOC | Self-organized criticality |

SOI | Southern Oscillation Index |

ULF | Ultra-low frequency |

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**Figure 1.**How a time series of (

**a**) dichotomous (e.g., zero or one) electric signals, (

**b**) earthquakes [18], (

**c**) avalanches in 3D rice piles [19,20], (

**d**) an ECG (for the so-called QT intervals), (

**e**) a not obviously dichotomous electric (or magnetic) signal and (

**f**) monthly Southern Oscillation Index (SOI) [21] values can be visualized in natural time. The meaning of the symbols, as well as the mathematical details for each case are discussed in Section 5.

**Figure 2.**Important properties of the entropy in natural time S and the entropy in natural time under time reversal ${S}_{-}$: in both panels, a signal consisting of 84 pulses is analyzed in natural time; the green and blue lines indicate the values (left scale) of S (green) and ${S}_{-}$ (blue) obtained for each $N(=10,11,12,\dots ,84)$. The signal is composed from 80 pulses of equal energy and four pulses, which are ten-times stronger (right scale, arbitrary units). In the upper panel, the stronger pulses are emitted periodically, while in the lower panel, consecutively in the middle of the process. Although the Shannon entropies for both panels are equal, the entropies in natural time S are different in each panel. Moreover, when using the entropy in natural time under time reversal ${S}_{-}$, we obtain values that are in general different from those of S.

**Figure 3.**Results from averaging the last 1000 events before a large avalanche ($s\ge $ 1000, occurring at ${T}_{0}$) in the Olami–Feder–Christensen (OFC) model with $L=$100 and $K=2$: the change $\mathrm{\Delta}S$ (left scale) of the entropy in natural time under time reversal and the mean energy $\zeta \equiv {\sum}_{i=1}^{L}{\sum}_{j=1}^{L}{z}_{ij}/{L}^{2}$ (right scale) as a function of the “time” (${T}_{0}-T$) to the large avalanche. Note that $\mathrm{\Delta}S$ minimizes before the occurrence time ${T}_{0}$ of the large avalanche and changes sign when $\zeta $ starts to increase.

**Figure 4.**The values (left scale) of ${\kappa}_{1}$ (red circles), S (blue lines) and ${S}_{-}$ (cyan lines) obtained from the study of seismicity within the region N${}_{36.0}^{38.6}$E${}_{20.0}^{22.5}$, estimated on 1 February 2008 by [121] that it will suffer a strong earthquake based on a seismic electric signal (SES) recorded on 14 January 2008 (see Figure 2(c) of [122]), after discarding the two events (earthquakes) that were related to another SES activity [122]. The black sticks correspond to the magnitude ${M}_{L}\left(ATH\right)$ (right scale) reported by the Geodynamical Institute of the National Observatory of Athens for each small earthquake. The horizontal red and blue lines correspond to 0.07 and ${S}_{u}$, respectively.

**Figure 5.**The entropy change $\mathrm{\Delta}{S}_{20}$ in natural time for the window length l = 20 months (red line, left scale) along with SOI monthly values (blue line, right scale) for the period January 2014–March 2016. The alarm is set on (black line) when $\mathrm{\Delta}{S}_{20}$ exceeds the threshold value $\mathrm{\Delta}{S}_{thres}=0.0035$ (red horizontal line). The selection of the threshold can be seen in Figure 2 of [127] and has been made as a compromise between the costs of false positive and false negative predictions. The two colored horizontal stripes represent the mean minimum negative values of SOI along with the one standard deviation bands for the two cases of “weak, weak to moderate, moderate, moderate to strong” (green band) and “strong, very strong” (yellow band) El Niño events [127].

**Figure 6.**PDF of $\mathrm{\Delta}{S}_{20}$ (blue curve, left scale) together with the corresponding histogram (red bars, left scale) obtained from the time series of $\mathrm{\Delta}{S}_{20}$, which is also plotted versus time (green crosses, right scale) along the vertical axis. The arrows indicate when $\mathrm{\Delta}{S}_{20}$ exceeds 0.0205 and are labeled by the corresponding ongoing strong El Niño events. Taken from [127].

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sarlis, N.V.
Entropy in Natural Time and the Associated Complexity Measures. *Entropy* **2017**, *19*, 177.
https://doi.org/10.3390/e19040177

**AMA Style**

Sarlis NV.
Entropy in Natural Time and the Associated Complexity Measures. *Entropy*. 2017; 19(4):177.
https://doi.org/10.3390/e19040177

**Chicago/Turabian Style**

Sarlis, Nicholas V.
2017. "Entropy in Natural Time and the Associated Complexity Measures" *Entropy* 19, no. 4: 177.
https://doi.org/10.3390/e19040177