# Research on Threshold Selection Method in Wave Extreme Value Analysis

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. POT Method

#### 2.2. Threshold Determination Using the Detrended Fluctuation Analysis Method

- (1)
- Determine the maximum ${x}_{\mathrm{max}}$ and minimum values ${x}_{\mathrm{min}}$ of the time series $\{x(t)\}$.
- (2)
- Determine the central point $R$ of the sequence $\{x(t)\}$, you can either take the average of all data points or choose a median value that lies between the maximum and minimum values.
- (3)
- Starting from the maximum value of $\{x(t)\}$, sequentially discard data points within intervals $\{x(t),\text{}x(t)\ge {x}_{\mathrm{max}}-d\times k\}$ until reaching the central point $R$. In this process, a series of new sequence ${Y}_{J},J={x}_{\mathrm{max}}-d\times k$ is obtained, where $d$ is the interval size.
- (4)
- Calculate the fractal exponent ${D}_{J}$ for each new sequence ${Y}_{J}$ and observe how it changes with the discarded interval size $J$.
- (5)
- When the change in ${D}_{J}$ starts to become smooth and converges to the original DFA exponent of the data $\{x(t)\}$, take the corresponding $J$ value as the threshold for extreme events in the sequence $\{x(t)\}$. The degree of convergence to the original value is not unique and may fluctuate slightly around the original exponent. Therefore, to determine the convergence point, the variance ${\mathrm{var}}_{j}$ of the sequence of exponents ${D}_{J}$ can be calculated. Variance can be defined as follows:$${\mathrm{var}}_{j}^{2}=\frac{1}{N-1}{\displaystyle \sum _{j=1}^{N}{(DF{A}_{j}-E)}^{2}}$$$$E=\frac{1}{N}{\displaystyle \sum _{j=1}^{N}DF{A}_{j}}$$

#### 2.3. Method Validation

#### 2.4. The Study Area and Data

## 3. Results

#### 3.1. The Long-Range Correlation of the Significant Wave Height Series

#### 3.2. Threshold Determination

#### 3.3. Return Period

#### 3.4. Comparison with Other Threshold Selection Methods

#### 3.4.1. The Mean Residual Life Plot

#### 3.4.2. Parameter Stability Plot

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The change in DFA exponent after excluding data from different intervals in a series with extreme values.

**Figure 2.**The change in DFA exponent after excluding data from different intervals in the original series.

**Figure 8.**Mean residual life plot. Lines represent empirical mean residual life plot and confidence intervals.

Location | Lat. (° N) | Lon. (° E) | Water Depth (m) | Maximum Wave Height (m) |
---|---|---|---|---|

P1 | 14 | 110 | 1274 | 9.75 |

P2 | 11 | 116 | 2671 | 5.74 |

P3 | 20 | 119 | 3032 | 12.65 |

P4 | 17 | 114 | 3053 | 9.55 |

P5 | 20 | 108 | 49 | 9.97 |

P6 | 21 | 116 | 128 | 10.54 |

Turning Point | n | ${\mathit{\chi}}^{2}$ | ${\mathit{\chi}}_{(\mathit{a}/2)}^{2}$ | ${\mathit{\chi}}_{(1-\mathit{a}/2)}^{2}$ | Significance Test |
---|---|---|---|---|---|

5.19 | 52 | 50.03 | 64.3 | 38.56 | No |

5.12 | 56 | 71.14 | 68.8 | 42.06 | Yes |

4.87 | 74 | 90.6 | 88.85 | 58.01 | Yes |

4.76 | 83 | 140.27 | 98.78 | 66.08 | Yes |

4.44 | 112 | 121.82 | 130.47 | 92.38 | No |

4.35 | 121 | 113.72 | 140.23 | 100.62 | No |

Return Periods | 3 Days | 4 Days | 5 Days | 6 Days | 7 Days |
---|---|---|---|---|---|

50 year (m) | 8.53 | 8.53 | 8.53 | 8.53 | 8.53 |

100 year (m) | 9.15 | 9.15 | 9.13 | 9.13 | 9.13 |

150 year (m) | 9.51 | 9.51 | 9.47 | 9.47 | 9.47 |

200 year (m) | 9.77 | 9.77 | 9.71 | 9.71 | 9.71 |

Locactions\Return Periods | 50 Year (m) | 100 Year (m) | 150 Year (m) | 200 Year (m) |
---|---|---|---|---|

P1 | 8.53 | 9.13 | 9.47 | 9.71 |

P2 | 6.17 | 6.24 | 6.28 | 6.31 |

P3 | 11.38 | 12.32 | 12.87 | 13.26 |

P4 | 9.84 | 10.42 | 10.75 | 10.99 |

P5 | 8.73 | 9.38 | 9.74 | 10.01 |

P6 | 10.67 | 11.42 | 11.85 | 12.14 |

Return Periods | Stable Threshold Range (m) | Return Period Significant Wave Heights (m) | The Average of Wave Height (m) |
---|---|---|---|

50 year | (4.60, 5.65) | (8.50, 8.55) | 8.53 |

100 year | (4.60, 5.55) | (9.09, 9.22) | 9.16 |

150 year | (4.60, 5.55) | (9.43, 9.61) | 9.52 |

200 year | (4.60, 5.55) | (9.65, 9.90) | 9.78 |

Return Periods | Stable Threshold Range (m) | Range of Differences (m) | Width of Differences (m) |
---|---|---|---|

50 year | (4.60, 5.65) | (−0.01, 0.09) | 0.10 |

100 year | (4.60, 5.55) | (−0.05, 0.06) | 0.11 |

150 year | (4.60, 5.55) | (−0.08, 0.10) | 0.18e |

200 year | (4.60, 5.55) | (−0.10, 0.14) | 0.24 |

Locations | P1 | P2 | P3 | P4 | P5 | P6 |
---|---|---|---|---|---|---|

Stable threshold range (m) | (4.65, 5.55) | (3.75, 4.95) | (5.90, 6.35) | (4.70, 5.70) | (4.10, 5.15) | (4.80, 5.55) |

Locations | MF-DFA (m) | Mean Excess Function (m) | Parameter Stability Plot (m) |
---|---|---|---|

P1 | 5.12 | (4.80, 5.40) | (4.85, 5.40) |

P2 | 3.96 | (3.90, 4.70) | (3.95, 4.40) |

P3 | 6.06 | (4.90, 6.50) | (5.80, 6.10) |

P4 | 5.59 | (4.70, 6.00) | (4.80, 5.60) |

P5 | 4.45 | (3.60, 4.50) | (4.10, 4.50) |

P6 | 5.15 | (4.98, 5.35) | (4.90, 5.20) |

Return Periods | MF-DFA (m) | Mean Excess Function (m) | Parameter Stability Plot (m) |
---|---|---|---|

50 year | 8.53 | 8.54 | 8.55 |

100 year | 9.13 | 9.19 | 9.22 |

150 year | 9.47 | 9.58 | 9.61 |

200 year | 9.71 | 9.85 | 9.89 |

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**MDPI and ACS Style**

Liu, H.; Yang, F.; Wang, H.
Research on Threshold Selection Method in Wave Extreme Value Analysis. *Water* **2023**, *15*, 3648.
https://doi.org/10.3390/w15203648

**AMA Style**

Liu H, Yang F, Wang H.
Research on Threshold Selection Method in Wave Extreme Value Analysis. *Water*. 2023; 15(20):3648.
https://doi.org/10.3390/w15203648

**Chicago/Turabian Style**

Liu, Huashuai, Fan Yang, and Hongchuan Wang.
2023. "Research on Threshold Selection Method in Wave Extreme Value Analysis" *Water* 15, no. 20: 3648.
https://doi.org/10.3390/w15203648