# Spurious Memory in Non-Equilibrium Stochastic Models of Imitative Behavior

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## Abstract

**:**

## 1. Introduction

## 2. Non-Equilibrium Stochastic Fluctuations Arising from the Imitative Behavior of Agents

## 3. PDFs of Burst and Inter-Burst Duration in the Stochastic Model of Imitative Behavior

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations

SDE | Stochastic differential equation |

fBm | fractional Brownian motion |

Probability density function | |

PSD | Power spectral density |

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

**a**), obtained by solving SDE (4) with ${\mu}_{i}$ given by Equation (6); (

**b**) PDF and (

**c**) PSD of the same time series (red curves). Black curves in (

**b**,

**c**) represent theoretical fits: (

**b**) beta distribution with parameter values two and two; (

**c**) $1/{f}^{2}$ trend line. Parameter set used in numerical simulation: $\alpha =2$, ${\epsilon}_{1}={\epsilon}_{2}=0$.

**Figure 2.**Excerpt of the transformed $y\left(t\right)=\frac{x\left(t\right)}{1-x\left(t\right)}$ time series (

**a**), where the $x\left(t\right)$ time series is the same as in Figure 1; (

**b**) PDF and (

**c**) PSD of the same transformed time series (red curves). Black curves in (

**b**,

**c**) represent theoretical fits: (

**b**) ${y}^{-3}$ trend line and (

**c**) $1/f$ trend line.

**Figure 3.**Excerpt of a generic time series. Three threshold, ${h}_{x}$, passage events, ${t}_{i}$, are shown. Thus burst duration T can be defined as $T={t}_{2}-{t}_{1}$, and inter-burst duration $\theta $ can be defined as $\theta ={t}_{3}-{t}_{2}$.

**Figure 4.**Comparison of PSD (

**a**,

**b**) and burst duration PDF (

**c**,

**d**) of the time series generated by numerically solving Equation (8) (

**a**,

**c**) with the ones obtained from fractional Brownian motion (fBm) with the relaxation time series, obtained by solving Equation (14) (

**b**,

**d**). fBm parameter sets: $\gamma =2$ (all cases), $H=0.1$ (red curve), $0.2$ (green curve), $0.3$ (blue curve), $0.4$ (magenta curve), $0.5$ (cyan curve). Parameters of Equation (8) are selected to give values of $H=(\beta -1)/2$ the same as for fBm: $\alpha =2$, $\epsilon =0.6$ (red curve), $1.2$ (green curve), $1.8$ (blue curve), $2.4$ (magenta curve), 3 (cyan curve).

**Figure 5.**PDFs of burst (red circles) and inter-burst (blue squares) durations of time series, obtained by solving SDE (4) with $\mu $ given by Equation (6), for various thresholds: ${h}_{x}=0.6$ (

**a**), $0.7$ (

**b**), $0.8$ (

**c**), $0.9$ (

**d**), $0.4$ (

**e**), $0.3$ (

**f**), $0.2$ (

**g**) and $0.1$ (

**h**). Parameter set used in numerical simulation: $\alpha =2$, ${\epsilon}_{1}={\epsilon}_{2}=0$. Solid black lines guide the eye according the power-law $3/2$.

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Gontis, V.; Kononovicius, A.
Spurious Memory in Non-Equilibrium Stochastic Models of Imitative Behavior. *Entropy* **2017**, *19*, 387.
https://doi.org/10.3390/e19080387

**AMA Style**

Gontis V, Kononovicius A.
Spurious Memory in Non-Equilibrium Stochastic Models of Imitative Behavior. *Entropy*. 2017; 19(8):387.
https://doi.org/10.3390/e19080387

**Chicago/Turabian Style**

Gontis, Vygintas, and Aleksejus Kononovicius.
2017. "Spurious Memory in Non-Equilibrium Stochastic Models of Imitative Behavior" *Entropy* 19, no. 8: 387.
https://doi.org/10.3390/e19080387