# Self-Similar Models: Relationship between the Diffusion Entropy Analysis, Detrended Fluctuation Analysis and Lévy Models

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## Abstract

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## 1. Introduction

**Definition**

**1**

**(Self-Similarity).**In the sense of finite dimensional distributions $\left(\stackrel{fdd}{=}\right)$, we define a stochastic process ${X}_{t}$ as self-similar if two processes ${\left\{{X}_{at}\right\}}_{t\ge 0}$ and ${\left\{{a}^{H}{X}_{t}\right\}}_{t\ge 0}$ have the same law if there exists $\mathit{H}>0$. i.e.,

## 2. Methods

#### 2.1. Lévy Flight Model

#### 2.2. Detrended Fluctuation Analysis (DFA)

#### 2.3. Diffusion Entropy Analysis (DEA)

- Select an integer l such that $1\le l\le M$.
- For each time, find $M-l+1$ sub-series of length l defined such that$${\u03f5}_{i}^{s}\equiv {\u03f5}_{i+s},i=1,\dots ,l$$
- For each sub-series, construct a distribution diffusion path by the position$${x}^{\left(s\right)}\left(l\right)=\sum _{i=1}^{l}{\u03f5}_{i}^{s}=\sum _{i=1}^{l}{\u03f5}_{i+s}$$
- To calculate the Shannon entropy of the diffusion process, partition the $x-axis$ into size $\eta \left(l\right)$ cells and tell how many parts there are in each of them at a time l. Let us call this number ${N}_{i}\left(l\right)$.
- With $N\left(l\right)$, we determine the probability ${p}_{i}\left(l\right)$ that a particle is found in the i-th cell at time l.$${p}_{i}\left(l\right)=\frac{{N}_{i}\left(l\right)}{M-l+1}$$
- The entropy of the diffusion process at time l is given by$${S}_{d}\left(l\right)=-\sum _{i}{p}_{i}\left(l\right)ln\left[{p}_{i}\left(l\right)\right]$$

#### 2.4. Background of Data

#### 2.4.1. Financial Market Data

#### 2.4.2. Geophysical Time Series Data

#### 2.4.3. Stationarity Test of Data

#### 2.4.4. Augmented Dickey-Fuller (ADF)

- ${H}_{0}:\gamma =0$
- ${H}_{a}:\gamma <0$

## 3. Results

#### 3.1. Numerical Results

#### Remarks on the Tables of Numerical Results

#### 3.2. Analytical Relations

**Property**

**1**

**.**Given a random variable X, if we denote with $Law\left(X\right)$ its probability density function (for example, for a Gaussian random variable, we write $Law\left(X\right)=N(\mu ,{\sigma}^{2})$) then we will say that the random variable X is stable, or that it has a stable distribution if for any $n\ge 2$ there exists a positive number ${C}_{n}$ and a number ${D}_{n}$ so that:

#### 3.2.1. Detrended Fluctuation Analysis (DFA)

#### 3.2.2. Diffusion Entropy Analysis (DEA)

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

- BLUE points depict the Normalized Data
- RED curve depicts the Cumulative Distribution Function
- GREEN curve depicts the Gaussian (Normal) distribution fit.

## References

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Financial Market | Test Statistic | Lag Order | p-Value |
---|---|---|---|

BVSP | −16.571 | 12 | 0.01 |

SPC | −19.992 | 13 | 0.01 |

HSI | −18.937 | 13 | 0.01 |

IGPA | −10.87 | 9 | 0.01 |

MERV | −13.298 | 10 | 0.01 |

MXX | −12.496 | 13 | 0.01 |

Nasdaq | −7.4649 | 10 | 0.01 |

PSI | −10.207 | 10 | 0.01 |

SETI | −8.4743 | 10 | 0.01 |

SP500 | −30.856 | 24 | 0.01 |

Station | Eruption | Test Statistic | Lag Order | p-Value |
---|---|---|---|---|

BEZB | 1 | −3.155 | 7 | 0.10 |

2 | −1.946 | 7 | 0.60 | |

BELO | 3 | −3.188 | 8 | 0.09 |

4 | −3.181 | 8 | 0.09 | |

5 | −3.342 | 8 | 0.07 | |

6 | −3.366 | 7 | 0.06 | |

7 | −2.784 | 7 | 0.25 | |

8 | −2.092 | 7 | 0.54 |

Markets | DFA (H) | DEA (δ) | Lévy (α) | H.α | δ.α |
---|---|---|---|---|---|

BVSP | 0.72 | 0.57 | 1.34 | 0.96 | 0.76 |

SPC | 0.62 | 0.60 | 1.40 | 0.87 | 0.84 |

HSI | 0.70 | 0.60 | 1.40 | 0.98 | 0.84 |

IGPA | 0.65 | 0.53 | 1.40 | 0.91 | 0.74 |

MERV | 0.62 | 0.56 | 1.40 | 0.87 | 0.78 |

MXX | 0.66 | 0.59 | 1.34 | 0.90 | 0.79 |

Nasdaq | 0.72 | 0.56 | 1.12 | 0.96 | 0.66 |

PSI | 0.71 | 0.55 | 1.40 | 0.99 | 0.77 |

SETI | 0.70 | 0.54 | 1.34 | 0.94 | 0.72 |

SP500 | 0.66 | 0.65 | 1.40 | 0.92 | 0.91 |

Seismic Station | Eruption Number | DFA (α) | DEA (δ) | Lévy (α) | H.α | δ.α |
---|---|---|---|---|---|---|

BEZB | 1 | 0.74 | 0.68 | 1.12 | 0.83 | 0.76 |

2 | 0.92 | 0.68 | 1.34 | 1.23 | 0.91 | |

BELO | 3 | 0.85 | 0.68 | 1.12 | 0.95 | 0.76 |

4 | 0.66 | 0.68 | 1.40 | 0.92 | 0.95 | |

5 | 0.76 | 0.68 | 1.12 | 0.85 | 0.76 | |

6 | 0.67 | 0.68 | 1.34 | 0.90 | 0.91 | |

7 | 0.81 | 0.68 | 1.40 | 1.13 | 0.95 | |

8 | 0.75 | 0.68 | 1.34 | 1.01 | 0.91 |

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**MDPI and ACS Style**

Mariani, M.C.; Kubin, W.; Asante, P.K.; Tweneboah, O.K.; Beccar-Varela, M.P.; Jaroszewicz, S.; Gonzalez-Huizar, H.
Self-Similar Models: Relationship between the Diffusion Entropy Analysis, Detrended Fluctuation Analysis and Lévy Models. *Mathematics* **2020**, *8*, 1046.
https://doi.org/10.3390/math8071046

**AMA Style**

Mariani MC, Kubin W, Asante PK, Tweneboah OK, Beccar-Varela MP, Jaroszewicz S, Gonzalez-Huizar H.
Self-Similar Models: Relationship between the Diffusion Entropy Analysis, Detrended Fluctuation Analysis and Lévy Models. *Mathematics*. 2020; 8(7):1046.
https://doi.org/10.3390/math8071046

**Chicago/Turabian Style**

Mariani, Maria C., William Kubin, Peter K. Asante, Osei K. Tweneboah, Maria P. Beccar-Varela, Sebastian Jaroszewicz, and Hector Gonzalez-Huizar.
2020. "Self-Similar Models: Relationship between the Diffusion Entropy Analysis, Detrended Fluctuation Analysis and Lévy Models" *Mathematics* 8, no. 7: 1046.
https://doi.org/10.3390/math8071046