# Wavelet-Based Multi-Scale Entropy Analysis of Complex Rainfall Time Series

## Abstract

**:**

## 1. Introduction

## 2. Wavelet Transform

#### 2.1. Introduction of Wavelet Transform

_{2}of the number of values in the input. This limitation is inappropriate for a hydrological time series, especially during the short duration of typhoon events in Taiwan.

#### 2.2. Redundant Wavelet Transform

## 3. Multi-Scale Entropy Analysis

#### 3.1. Sample Entropy

#### 3.2. Multi-Scale Entropy

## 4. Application and Analysis

**Figure 1.**The locations of the four rainfall stations [16].

## 5. Results and Discussion

**Table 1.**The entropy measures of the original signal (S0) at various scale factors for four stations.

Scale Factor | 01B030 | 01J930 | 01P660 | 01T070 |
---|---|---|---|---|

1 | 1.44280 | 0.87499 | 0.64722 | 1.22523 |

2 | 1.10868 | 1.30938 | 1.21979 | 1.23958 |

3 | 0.91505 | 0.96548 | 1.15350 | 0.96414 |

4 | 0.80653 | 0.81377 | 0.85012 | 0.76989 |

5 | 0.72637 | 0.75014 | 0.62295 | 0.57343 |

6 | 0.64986 | 0.36981 | 0.42154 | 0.41224 |

7 | 0.47673 | 0.29079 | 0.31590 | 0.34716 |

8 | 0.44270 | 0.36412 | 0.34077 | 0.31724 |

9 | 0.19843 | 0.29225 | 0.15761 | 0.16510 |

10 | 0.36687 | 0.06402 | 0.31995 | 0.29353 |

Average | 0.71340 | 0.60948 | 0.60494 | 0.63075 |

**Table 2.**The standard normal variate of different MSE curves for four stations (m = 2 and r = 0.15SD).

S0 | S1 | S2 | S3 | S4 | S5 | Suggested Level | |
---|---|---|---|---|---|---|---|

01B030 | −3.846 | −3.309 | −3.309 | −0.447 | 2.952 | 3.488 | 3 |

01J930 | −3.309 | −2.773 | −2.952 | −1.699 | 2.952 | 4.025 | 3 |

01P660 | −2.952 | −2.415 | −2.952 | −2.415 | −0.626 | 4.025 | 4 |

01T070 | −3.667 | −3.667 | −3.130 | −2.415 | 1.342 | 2.952 | 4 |

**Table 3.**The standard normal variate of different MSE curves at four stations when removing 25% of the time series.

S0 | S1 | S2 | S3 | S4 | S5 | Suggested Level | |
---|---|---|---|---|---|---|---|

01B030 | −3.488 | −3.667 | −3.130 | −2.057 | 3.130 | 3.130 | 3 |

01J930 | −3.130 | −3.309 | −2.773 | −1.521 | 3.488 | 4.025 | 3 |

01P660 | −2.773 | −2.594 | −2.773 | −2.415 | 0.626 | 2.594 | 4 |

01T070 | −3.667 | −3.130 | −3.667 | −2.415 | 2.594 | 2.236 | 3 |

S0 | S1 | S2 | S3 | S4 | S5 | Suggested Level | |
---|---|---|---|---|---|---|---|

01B030 | −3.488 | −3.488 | −2.415 | 0.268 | 3.667 | 3.846 | 3 |

01J930 | −1.163 | −1.342 | 0.089 | 0.805 | 2.594 | 4.025 | 3 |

01P660 | −0.268 | −0.089 | 0.805 | 0.447 | 1.521 | 3.846 | 4 |

01T070 | −2.952 | −2.415 | −1.699 | −1.878 | 1.699 | 4.025 | 4 |

**Table 5.**The standard normal variate of different MSE curves at the four stations (m = 2 and r = 0.20SD).

S0 | S1 | S2 | S3 | S4 | S5 | Suggested Level | |
---|---|---|---|---|---|---|---|

01B030 | −3.846 | −3.130 | −2.057 | 0.805 | 3.309 | 3.846 | 3 |

01J930 | −2.952 | −2.594 | −0.805 | 0.089 | 4.025 | 4.025 | 3 |

01P660 | −2.594 | −2.236 | −1.878 | −1.342 | 2.594 | 3.846 | 3 |

01T070 | −3.488 | −3.309 | −2.952 | −0.626 | 2.594 | 3.846 | 3 |

**Table 6.**The standard normal variate of different MSE curves at the four stations (m = 3 and r = 0.25SD).

S0 | S1 | S2 | S3 | S4 | S5 | Suggested Level | |
---|---|---|---|---|---|---|---|

01B030 | −3.667 | −2.415 | −3.309 | −0.626 | 3.488 | 3.667 | 3 |

01J930 | −2.952 | −2.952 | −2.952 | −1.699 | 4.025 | 4.025 | 3 |

01P660 | −3.130 | −2.773 | −2.594 | −2.594 | 1.699 | 3.846 | 4 |

01T070 | −2.952 | −3.488 | −3.309 | −2.236 | 0.984 | 4.025 | 4 |

## 6. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Chou, C.-M. Wavelet-Based Multi-Scale Entropy Analysis of Complex Rainfall Time Series. *Entropy* **2011**, *13*, 241-253.
https://doi.org/10.3390/e13010241

**AMA Style**

Chou C-M. Wavelet-Based Multi-Scale Entropy Analysis of Complex Rainfall Time Series. *Entropy*. 2011; 13(1):241-253.
https://doi.org/10.3390/e13010241

**Chicago/Turabian Style**

Chou, Chien-Ming. 2011. "Wavelet-Based Multi-Scale Entropy Analysis of Complex Rainfall Time Series" *Entropy* 13, no. 1: 241-253.
https://doi.org/10.3390/e13010241