# Entropic Approach to the Detection of Crucial Events

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## Abstract

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## 1. Introduction

## 2. Pink Noise

#### 2.1. Stationary Fractional Brownian Motion

#### 2.2. Aging Fractional Brownian Motion

## 3. Entropy Concept

#### 3.1. External Entropy

#### 3.2. Entropic Treatment of the Scale Detection Issue

## 4. Refined Diffusion Entropy Analysis

#### 4.1. Without Stripes

#### 4.2. With Stripes

## 5. Details on the Action of the Stripes

## 6. Criticality and Physiological Processes

## 7. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FBM | Fractional Brownian Motion |

SFBM | Stationary Fractional Brownian Motion |

AFBM | Aging Fractional Brownian Motion |

DEA | Diffusion Entropy Analysis |

Probability Distribution Function | |

IPL | Inverse Power Law |

SOTC | Self-Organized Temporal Criticality |

EEG | Electroencephalogram |

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**Figure 1.**Effect of embedding crucial events in a dense cloud of Poisson events.

**Left panel:**Correlation function of the time series of Equation (51), $\u03f5=0.1$, $\Gamma =1$. $\lambda =1.6$, $\mu =2.6$;

**Right panel:**DEA without stripes. The slope of the intermediate asymptotics fits the prediction of Equation (43).

**Figure 2.**Comparison between the use of DEA with stripes, left panel, and the use of DEA without stripes, right panel.

**Figure 3.**Effect of embedding crucial events in a dense cloud of Poisson events compared to the effect of embedding them into a cloud of SFBM fluctuations. In both panels $\u03f5=0.1$, $\lambda =1.6$, $\mu =2.6$;

**Left panel:**$\Gamma =1$.

**Right panel:**$H=0.7.$

**Figure 5.**This figure illustrates the signal $\xi \left(t\right)$ used as surrogate sequence for the method of DEA illustrated in Figure 3.

**Figure 6.**The top panel illustrates the fluctuation $\xi \left(t\right)$ of Equation (56). The bottom illustrates the jumps $z\left(t\right)$ done by the random walker to generate the trajectory $x\left(t\right)$ of Equation (54). The random walker makes a jump ahead of intensity 1 when the fluctuation $\xi \left(t\right)$ of the top panel crosses the border lines between consecutive stripes.

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Culbreth, G.; West, B.J.; Grigolini, P.
Entropic Approach to the Detection of Crucial Events. *Entropy* **2019**, *21*, 178.
https://doi.org/10.3390/e21020178

**AMA Style**

Culbreth G, West BJ, Grigolini P.
Entropic Approach to the Detection of Crucial Events. *Entropy*. 2019; 21(2):178.
https://doi.org/10.3390/e21020178

**Chicago/Turabian Style**

Culbreth, Garland, Bruce J. West, and Paolo Grigolini.
2019. "Entropic Approach to the Detection of Crucial Events" *Entropy* 21, no. 2: 178.
https://doi.org/10.3390/e21020178