# The Multiscale Fluctuations of the Correlation between Oil Price and Wind Energy Stock

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## Abstract

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## 1. Introduction

## 2. Data and Algorithm Description

#### 2.1. Data Description

#### 2.2. Algorithms

#### 2.2.1. Co-Movement in Time Domain

#### 2.2.2. Decomposition into the Time–Frequency Domain

#### 2.2.3. Construction of the Multiscale Conformation Evolution Network

- Step 1.
- The discretization of the wavelet power spectrum. The wavelet power spectrum shows a clearer picture of the fluctuation of the correlation of Brent–wind in detail. The frequency band of the wavelet power spectrum is continuous and rings from 2 to 512 days. Here, we choose the frequency band of 2 days, 4 days, 8 days, 16 days, 32 days, 64 days, 128 days, 256 days, and 512 days as the backbone to stand for the entire continuous frequency band according to the discretization method of the discrete wavelet transform. As mentioned, the wavelet power spectrum is a n*m matrix, where n is the number of frequency bands that constitute the continuous frequency band, and m is the number of data points. After the discretization we obtain a new matrix, the size of which is 9*m. We normalize the discretized wavelet power spectrum matrix through the logarithmic transform.
- Step 2.
- The definition of the multiscale conformation. First, we divide the values of the discretized wavelet power spectrum into four levels according to its maximum and minimum and divide the value ranges from the minimum to the maximum into four equal intervals. We also use L1, L2, L3, and L4, which mean the fluctuation is very weak, weak, high, and very high, respectively. For each time point, there are nine components from different time horizons, and the combination of them can determine the status of the original Brent–wind correlation. Here, we define the multiscale combination of nine time scales at one time point as the multiscale conformation.
- Step 3.
- Constructing an evolution network for the multiscale conformations. Taking multiscale conformations as nodes, transmissions among them as edges, we transform the series of multiscale conformations into a network. Details are shown in Figure 2. Among nodes and edges, transmissions between two types of multiscale conformations appear repetitively; therefore, we take the n umber at which these edges appear as its weight.

## 3. Empirical Results and Discussion

#### 3.1. Fluctuations of Correlations of Brent–wind in the Time Domain

#### 3.2. Decomposition in Time–Frequency Domain

#### 3.3. Construction of the Evolution Network

#### 3.4. Evolution Features Analyses

#### 3.4.1. Major Conformations and Transmissions

_{i}represents the number of neighbors shared by the conformation ${w}_{i}$, and ${w}_{j}$ represents the weight between conformation ${w}_{i}$ and ${w}_{j}$. Figure 5 depicts features of the weighted out-degree and we find an 80.12% transmission occur among 31.85% of the total conformations (Figure 7a). In addition, the distribution of the weighted out-degree follows the low power distribution $p({w}^{out})~{w}^{out-\gamma}$ (Figure 7b), which also demonstrates that a small amount of multiscale conformations is more vital than others.

#### 3.4.2. The Clustering Effect

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Original prices and returns of the Brent oil price and wind energy stock index. (

**a**) plots the original price of Brent oil price and the wind energy stock index, and (

**b**) indicates the returns of Brent oil price and the wind energy stock index.

**Figure 7.**Distributions of the weighted out-degree of node. (

**a**) Cumulative distribution of the weighted out-degree of the node (sorted by the value of the weighted out-degree of the nodes in descending order, n = 1, 2, . . ., 270). (

**b**) Double-logarithmic plot between the weighted out-degree of node and the probability of the weighted out-degree.

No. | Conformations | Weighted Out-Degree | Percentage (%) Accounts for Total Weighted Out-Degree |
---|---|---|---|

1 | L_{3}L_{3}L_{3}L_{3}L_{3}L_{3}L_{4}L_{4}L_{4} | 649 | 4.69 |

2 | L_{3}L_{3}L_{3}L_{4}L_{3}L_{3}L_{4}L_{4}L_{4} | 291 | 4.37 |

3 | L_{3}L_{3}L_{3}L_{3}L_{3}L_{4}L_{4}L_{4}L_{4} | 228 | 3.31 |

4 | L_{3}L_{3}L_{4}L_{4}L_{3}L_{4}L_{4}L_{4}L_{4} | 152 | 3.22 |

5 | L_{3}L_{3}L_{3}L_{4}L_{3}L_{4}L_{4}L_{4}L_{4} | 151 | 2.82 |

6 | L_{3}L_{3}L_{4}L_{4}L_{3}L_{3}L_{4}L_{4}L_{4} | 149 | 2.57 |

7 | L_{3}L_{3}L_{3}L_{3}L_{4}L_{4}L_{4}L_{4}L_{4} | 115 | 2.20 |

8 | L_{3}L_{4}L_{3}L_{3}L_{3}L_{3}L_{4}L_{4}L_{4} | 101 | 2.20 |

9 | L_{3}L_{3}L_{4}L_{3}L_{3}L_{4}L_{4}L_{4}L_{4} | 99 | 2.12 |

10 | L_{3}L_{3}L_{4}L_{3}L_{3}L_{3}L_{4}L_{4}L_{4} | 99 | 1.96 |

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**MDPI and ACS Style**

Huang, S.; An, H.; Gao, X.; Jiang, M. The Multiscale Fluctuations of the Correlation between Oil Price and Wind Energy Stock. *Sustainability* **2016**, *8*, 534.
https://doi.org/10.3390/su8060534

**AMA Style**

Huang S, An H, Gao X, Jiang M. The Multiscale Fluctuations of the Correlation between Oil Price and Wind Energy Stock. *Sustainability*. 2016; 8(6):534.
https://doi.org/10.3390/su8060534

**Chicago/Turabian Style**

Huang, Shupei, Haizhong An, Xiangyun Gao, and Meihui Jiang. 2016. "The Multiscale Fluctuations of the Correlation between Oil Price and Wind Energy Stock" *Sustainability* 8, no. 6: 534.
https://doi.org/10.3390/su8060534