# Long-Range Correlations and Characterization of Financial and Volcanic Time Series

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## Abstract

**:**

## 1. Introduction

## 2. Variance Scaling Methods

#### 2.1. Rescaled Range Analysis

#### 2.2. Detrended Fluctuation Analysis

#### 2.3. Diffusion Entropy Analysis

#### 2.4. Estimation Procedure

#### The Shannon Entropy

- The time series data is first transformed into a diffusion process.
- Shannon’s entropy of the diffusion process is calculated. A log-linear equation or log-quadratic equation is derived from the Shannon entropy by substituting Equations (3) and (4) respectively. Simplifying the result from the substitutions, we have the following relation for stationary time series:$$S\left(t\right)=A+\delta ln\left(t\right).$$For the non-stationary series, the relation is as follows:$$S\left(t\right)=A+\delta \left(t\right)\tau ,$$$$S\left(t\right)=A+({\delta}_{0}-K)log\left(t\right)+(1-{\delta}_{0}){(log\left(t\right))}^{2},$$Thus $\delta $ (or ${\delta}_{0})$ is derived by an estimation of the slope of the above linear-log equation or by the coefficients from the quadratic-log equation. For details of the algorithm used when transforming the series into a diffusion process, we refer the reader to Reference [7].

## 3. Financial and Volcanic Time Series

#### 3.1. Financial Time Series

#### 3.2. Volcanic Time Series

#### 3.3. Stationarity of the Financial and Volcanic Time Series

#### 3.3.1. Augmented Dickey-Fuller

#### 3.3.2. Financial Time Series

#### 3.3.3. Volcanic Time Series

## 4. Results

#### Figures

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The hourly occupancy rate of a road in the bay area for 2 weeks [6].

Market | p-Value |
---|---|

BVSP | 0.015 |

SPC | 0.034 |

HSI | 0.033 |

IGPA | 0.03 |

MERV | 0.014 |

MXX | 0.024 |

Nasdaq | 0.04 |

PSI | <0.01 |

SETI | <0.01 |

SP500 | <0.01 |

XU100 | 0.01 |

Eruption Number | p-Value |
---|---|

1 | 0.3568 |

2 | 0.6747 |

3 | 0.3024 |

4 | 0.095 |

5 | 0.2064 |

6 | 0.3271 |

7 | 0.2374 |

8 | 0.4059 |

Market | R/S(H) | DFA ($\mathit{\alpha}$) | DEA ($\mathit{\delta}$) | ${\mathit{\delta}}_{\mathit{Levy}}$ (R/S) | ${\mathit{\delta}}_{\mathit{Levy}}$ (DFA) |
---|---|---|---|---|---|

BVSP | 0.59 | 0.72 | 0.57 | 0.56 | 0.63 |

SPC | 0.59 | 0.62 | 0.60 | 0.56 | 0.56 |

HSI | 0.65 | 0.7 | 0.60 | 0.56 | 0.63 |

IGPA | 0.74 | 0.65 | 0.53 | 0.63 | 0.56 |

MERV | 0.62 | 0.62 | 0.56 | 0.56 | 0.56 |

MXX | 0.64 | 0.66 | 0.59 | 0.56 | 0.56 |

Nasdaq | 0.6 | 0.72 | 0.56 | 0.56 | 0.56 |

PSI | 0.66 | 0.71 | 0.55 | 0.63 | 0.56 |

SETI | 0.64 | 0.70 | 0.54 | 0.56 | 0.56 |

SP500 | 0.63 | 0.66 | 0.65 | 0.58 | 0.60 |

XU100 | 0.64 | 0.70 | 0.54 | 0.56 | 0.56 |

Eruption Number | R/S(H) | DFA ($\mathit{\alpha}$) | DEA ($\mathit{\delta}$) | ${\mathit{\delta}}_{\mathit{Levy}}$ (R/S) | ${\mathit{\delta}}_{\mathit{Levy}}$ (DFA) |
---|---|---|---|---|---|

1 | 0.45 | 0.74 | 0.6837 | 0.4756 | 0.6547 |

2 | 0.51 | 0.92 | 0.6837 | 0.5093 | 0.8682 |

3 | 0.38 | 0.85 | 0.6837 | 0.4472 | 0.7636 |

4 | 0.39 | 0.66 | 0.6837 | 0.4509 | 0.5957 |

5 | 0.39 | 0.76 | 0.6837 | 0.4513 | 0.6729 |

6 | 0.37 | 0.67 | 0.6837 | 0.4433 | 0.6002 |

7 | 0.42 | 0.81 | 0.6837 | 0.4634 | 0.7194 |

8 | 0.504 | 0.75 | 0.6837 | 0.5018 | 0.6684 |

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**MDPI and ACS Style**

Mariani, M.C.; Asante, P.K.; Bhuiyan, M.A.M.; Beccar-Varela, M.P.; Jaroszewicz, S.; Tweneboah, O.K.
Long-Range Correlations and Characterization of Financial and Volcanic Time Series. *Mathematics* **2020**, *8*, 441.
https://doi.org/10.3390/math8030441

**AMA Style**

Mariani MC, Asante PK, Bhuiyan MAM, Beccar-Varela MP, Jaroszewicz S, Tweneboah OK.
Long-Range Correlations and Characterization of Financial and Volcanic Time Series. *Mathematics*. 2020; 8(3):441.
https://doi.org/10.3390/math8030441

**Chicago/Turabian Style**

Mariani, Maria C., Peter K. Asante, Md Al Masum Bhuiyan, Maria P. Beccar-Varela, Sebastian Jaroszewicz, and Osei K. Tweneboah.
2020. "Long-Range Correlations and Characterization of Financial and Volcanic Time Series" *Mathematics* 8, no. 3: 441.
https://doi.org/10.3390/math8030441