# Extreme Precipitation Frequency Analysis Using a Minimum Density Power Divergence Estimator

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Decadal Changes in Extreme Rainfall in South Korea

#### 2.2. MDPDE

## 3. Results and Discussion

#### 3.1. Decadal Changes in Extreme Rainfall in South Korea

#### 3.2. Physical Mechanism Behind the Changes of Extreme Events

#### 3.3. Performance of MDPDE with the Gumbel Distribution

_{2}distance estimator, respectively. The MDPDE for the Gumbel distribution is summarized in Table 3, where $\hat{\mathsf{\mu}}$ and $\hat{\mathsf{\sigma}}$ represent the estimates for location and scale parameters, respectively. As shown in Figure 6, MLE (α = 0) tends to be affected more by the extreme event (870.5 mm) than the L

_{2}distance estimator (α = 1), which leads to more inclined distributions to the right or to the extreme event as the value of α decreases. As a result, the design rainfalls obtained with MLE (α = 0) are highest compared to other values of α regardless of the return periods. The difference between the design rainfall estimated with MLE and the L

_{2}distance estimator was up to 17.3% depending on the recurrence interval. Consequently, the result shows that the MDPDE with α greater than zero improves the robustness, and hence, decrease the impact of an extreme event compared to MLE (α = 0).

#### 3.4. Balance between Robustness and Asymptotic Efficiency Based on the Optimal Value of $\mathsf{\alpha}$

_{2}distance estimator by 11.2%–11.5% and also less than those from the MLE by 6.3%–7.7% depending on the recurrence interval from 10 to 300 years. These results indicate that MDPDE with an optimal value of α suggests an alternative way to evaluate the design rainfall with the balance between the asymptotic efficiency and robustness.

#### 3.5. Performance of the MDPDE with GEV Depending on the Magnitude of the Extremes

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Park, S.K.; Lee, E. Synoptic features of orographically enhanced heavy rainfall on the east coast of Korea associated with typhoon Rusa (2002). Geophys. Res. Lett.
**2007**, 34. [Google Scholar] [CrossRef] - Lee, D.K.; Choi, S.J. Observation and numerical prediction of torrential rainfall over Korea caused by typhoon Rusa (2002). J. Geophys. Res. Atmos.
**2010**, 115. [Google Scholar] [CrossRef] - Kim, N.W.; Won, Y.S.; Chung, I.M. The scale of typhoon Rusa. Hydrol. Earth Syst. Sci. Discuss.
**2006**, 3, 3147–3182. [Google Scholar] [CrossRef] - Franks, S.W.; Kuczera, G. Flood frequency analysis: Evidence and implications of secular climate variability, New South Wales. Water Resour. Res.
**2002**. [Google Scholar] [CrossRef] - Alexander, L.V.; Zhang, X.; Peterson, T.C.; Caesar, J.; Gleason, B.; Tank, A.M.G.K.; Haylock, M.; Collins, D.; Trewin, B.; Rahimzadeh, F.; et al. Global observed changes in daily climate extremes of temperature and precipitation. J. Geophys. Res. Atmos.
**2006**, 111. [Google Scholar] [CrossRef] - Madsen, H.; Lawrence, D.; Lang, M.; Martinkova, M.; Kjeldsen, T.R. Review of trend analysis and climate change projections of extreme precipitation and floods in Europe. J. Hydrol.
**2014**, 519, 3634–3650. [Google Scholar] [CrossRef] [Green Version] - Trenberth, K.E. Changes in precipitation with climate change. Clim. Res.
**2011**, 47, 123–138. [Google Scholar] [CrossRef] - Intergovernmental Panel on Climate Change (IPCC). Climate Change 2014: Synthesis Report. Contribution of Working Groups I, II and III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change; IPCC: Geneva, Switzerland, 2014; p. 151.
- Prosdocimi, I.; Kjeldsen, T.R.; Miller, J.D. Detection and attribution of urbanization effect on flood extremes using nonstationary flood-frequency models. Water Resour. Res.
**2015**, 51, 4244–4262. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Coles, S.; Pericchi, L.R.; Sisson, S. A fully probabilistic approach to extreme rainfall modeling. J. Hydrol.
**2003**, 273, 35–50. [Google Scholar] [CrossRef] - Strupezewski, W.G.; Kochanek, K.; Weglarczyk, S.; Singh, V.P. On robustness of large quantile estimates of log-Gumbel and log-logistic distributions to largest element of the observation series: Monte Carlo results vs. first order approximation. Stoch. Environ. Res. Risk Assess.
**2005**, 19, 280–291. [Google Scholar] [CrossRef] - Strupczewski, W.G.; Kochanek, K.; Weglarczyk, S.; Singh, V.P. On robustness of large quantile estimates to largest elements of the observation series. Hydrol. Process.
**2007**, 21, 1328–1344. [Google Scholar] [CrossRef] - Cunnane, C. Factors affecting choice of distribution for flood series. Hydrol. Sci. J.
**1985**, 30, 25–36. [Google Scholar] [CrossRef] - Mutua, F.M. The use of the akaike information criterion in the identification of an optimum flood frequency model. Hydrol. Sci. J.
**1994**, 39, 235–244. [Google Scholar] [CrossRef] - Neykov, N.; Filzmoser, P.; Dimova, R.; Neytchev, P. Robust fitting of mixtures using the trimmed likelihood estimator. Comput. Stat. Data Anal.
**2007**, 52, 299–308. [Google Scholar] [CrossRef] - Singh, V.P. Hydrologic Frequency Modeling. In Proceedings of the International Symposium on Flood Frequency and Risk Analyses, Baton Rouge, LA, USA, 14–17 May 1986; Springer Netherlands: Dordrecht, The Netherlands, 1987; p. 645. [Google Scholar]
- Smith, R.L. Extreme value theory based on the r largest annual events. J. Hydrol.
**1986**, 86, 27–43. [Google Scholar] [CrossRef] - Strupczewski, W.G.; Kochanek, K.; Singh, V.P. On the informative value of the largest sample element of log-gumbel distribution. Acta Geophys.
**2007**, 55, 652–678. [Google Scholar] [CrossRef] - Basu, A.; Harris, I.R.; Hjort, N.L.; Jones, M.C. Robust and efficient estimation by minimising a density power divergence. Biometrika
**1998**, 85, 549–559. [Google Scholar] [CrossRef] - Beran, R. Minimum hellinger distance estimates for parametric models. Ann. Stat.
**1977**, 5, 445–463. [Google Scholar] [CrossRef] - Tamura, R.N.; Boos, D.D. Minimum hellinger distance estimation for multivariate location and covariance. J. Am. Stat. Assoc.
**1986**, 81, 223–229. [Google Scholar] [CrossRef] - Simpson, D.G. Minimum hellinger distance estimation for the analysis of count data. J. Am. Stat. Assoc.
**1987**, 82, 802–807. [Google Scholar] [CrossRef] - Basu, A.; Lindsay, B. Minimum disparity estimation for continuous models: Efficiency, distributions and robustness. Ann. Inst. Stat. Math.
**1994**, 46, 683–705. [Google Scholar] [CrossRef] - Cao, R.; Cuevas, A.; Fraiman, R. Minimum distance density-based estimation. Comput. Stat. Data Anal.
**1995**, 20, 611–631. [Google Scholar] [CrossRef] - Mihoko, M.; Eguchi, S. Robust blind source separation by beta divergence. Neural Comput.
**2002**, 14, 1859–1886. [Google Scholar] [CrossRef] [PubMed] - Lee, S.; Song, J. Minimum density power divergence estimator for garch models. Test
**2009**, 18, 316–341. [Google Scholar] [CrossRef] - Kim, B.; Lee, S. Robust estimation for the covariance matrix of multivariate time series based on normal mixtures. Comput. Stat. Data Anal.
**2013**, 57, 125–140. [Google Scholar] [CrossRef] - Karl, T.R.; Knight, R.W.; Easterling, D.R.; Quayle, R.G. Indices of climate change for the United States. Bull. Am. Meteorol. Soc.
**1996**, 77, 279–292. [Google Scholar] [CrossRef] - Groisman, P.Y.; Karl, T.R.; Easterling, D.R.; Knight, R.W.; Jamason, P.F.; Hennessy, K.J.; Suppiah, R.; Page, C.M.; Wibig, J.; Fortuniak, K.; et al. Changes in the probability of heavy precipitation: Important indicators of climatic change. Clim. Chang.
**1999**, 42, 243–283. [Google Scholar] [CrossRef] - Kunkel, K.E.; Andsager, K.; Easterling, D.R. Long-term trends in extreme precipitation events over the conterminous United States and Canada. J. Clim.
**1999**, 12, 2515–2527. [Google Scholar] [CrossRef] - Groisman, P.Y.; Knight, R.W.; Karl, T.R. Heavy precipitation and high streamflow in the contiguous United States: Trends in the twentieth century. Bull. Am. Meteorol. Soc.
**2001**, 82, 219–246. [Google Scholar] [CrossRef] - Chu, P.S.; Chen, Y.R.; Schroeder, T.A. Changes in precipitation extremes in the Hawaiian Islands in a warming climate. J. Clim.
**2010**, 23, 4881–4900. [Google Scholar] [CrossRef] - Spierre, S.G.; Wake, C. Trends in Extreme Precipitation Events for the Northeastern United States 1948–2007; University of New Hampshire: Durham, NH, USA, 2010. [Google Scholar]
- Korea Meteorological Administration (KMA). Open Portal for Meteorological Data in South Korea. Available online: https://data.kma.go.kr/cmmn/main.do (accessed on 10 September 2016).
- Yen, B.C.; Tung, Y.-K.; American Society of Civil Engineers. Subcommittee on Uncertainty and Reliability Analysis in Design of Hydraulic Structures. In Reliability and Uncertainty Analyses in Hydraulic Design: A Report; American Society of Civil Engineers: New York, NY, USA, 1993; p. 291. [Google Scholar]
- Korea Meteorological Administration (KMA). Fact Book of Climate Change in Korea; Korea Meteorological Administration: Seoul, Korea, 2011; p. 117.
- Korea Meteorological Administration (KMA). Learning from Recent Events in 20 Years: Top 10 Severe Rainfall Events; Korea Meteorological Administration: Seoul, Korea, 2012; p. 48.
- Choi, K.-S.; Park, K.-J.; Kim, J.-Y.; Kim, B.-J. Synoptic analysis on the trend of northward movement of tropical cyclone with maximum intensity. J. Korean Earth Sci. Soc.
**2015**, 36, 171–180. [Google Scholar] [CrossRef] - Choi, K.S.; Moon, I.J. Changes in tropical cyclone activity that has affected Korea since 1999. Nat. Hazards
**2012**, 62, 971–989. [Google Scholar] [CrossRef] - Oh, S.M.; Moon, I.J. Typhoon and storm surge intensity changes in a warming climate around the Korean peninsula. Nat. Hazards
**2013**, 66, 1405–1429. [Google Scholar] [CrossRef] - Fujisawa, H.; Eguchi, S. Robust estimation in the normal mixture model. J. Stat. Plan. Inference
**2006**, 136, 3989–4011. [Google Scholar] [CrossRef]

**Figure 2.**Changes in extreme rainfall: (

**a**) percent changes in number of events greater than 50.8 mm (2 in); (

**b**) percent changes in number of events greater than 101.6 mm (4 in); (

**c**) percent changes in volume of events greater than the 90th percentile; and (

**d**) percent changes in volume of events greater than the 99th percentile.

**Figure 3.**(

**a**) Changes in the volume of 24-h rainfall with a 100-year return period; and (

**b**) changes in the design rainfall for Seoul (Station 108) depending on various return periods.

**Figure 4.**Boxplots for the annual maxima during: (

**a**) 1974–1994; and (

**b**) 1995–2014 for the 24-h duration; the grey dash shows the mean of the outliers during 1974–1994.

**Figure 5.**(

**a**) Changes in the thresholds that determine the outliers for the annual maxima of 60 stations with a rainfall duration of 24 h; (

**b**) number of outliers during 1974–1994 and 1995–2014; and (

**c**) mean of the outliers for each period depending on the rainfall durations.

**Figure 6.**Gumbel distribution fitted to the 24-h annual maxima at Gangneung (Station 105) and design rainfall depending on the return periods.

**Figure 7.**Gumbel distribution with the optimal tuning parameter (α = 0.092) and the corresponding design rainfall depending on the return periods.

**Figure 8.**Estimated design rainfall from the GEV fitted to 24-h annual maxima at Gangneung (Station 105).

**Figure 9.**(

**a**) GEV (0, 1, 0.1) with no outlier; (

**b**) replacing the maximum with an outlier of 8; and (

**c**) with an outlier of 10.

**Figure 10.**(

**a**) GEV (0, 1, 0.2) replacing maximum with an outlier of 8; (

**b**) with an outlier of 10; and (

**c**) with outlier of 25.

Methods | Description |
---|---|

Actual rainfall amounts | Heavy rainfall: an event with precipitation above 50.8 mm (2 in) Very heavy rainfall: an event with precipitation above 101.6 mm (4 in) |

Specific thresholds | Heavy rainfall: an event with precipitation above 90th percentiles Very heavy rainfall: an event with precipitation above 99th percentiles |

Return periods | Heavy rainfall event: an event with 24-h precipitation above the 20 year return period Very heavy rainfall: an event with 24-h precipitation above the 100 year return period |

No. | Province | Abbreviation | Rain Gauges |
---|---|---|---|

1 | Seoul Metropolitan City | SM | 108 |

2 | Incheon Metropolitan City | IM | 112 |

3 | Gyeonggi-do | GG | 119, 201, 202, 203 |

4 | Gangwon-do | GW | 90, 100, 101, 105, 114, 211, 212 |

5 | Chungcheongbuk-do | CB | 127, 131, 226 |

6 | Chungcheongnam-do | CN | 129, 135, 140, 232, 235, 236 |

7 | Daejeon Metropolitan City | DJM | 133 |

8 | Gyeongsangbuk-do | GB | 115, 130, 138, 272, 273, 277, 279, 281 |

9 | Gyeongsangnam-do | GN | 162, 192, 284, 288, 289, 295 |

10 | Daegu Metropolitan City | DGM | 143 |

11 | Ulsan Metropolitan City | UM | 152 |

12 | Busan Metropolitan City | BM | 159 |

13 | Jeolabuk-do | JB | 146, 243, 244, 245, 247 |

14 | Jeolanam-do | JN | 165, 170, 260, 261, 262 |

15 | Gwangju Metropolitan City | GM | 156 |

16 | Jeju Island | JS | 184, 188, 189 |

$\mathsf{\alpha}$ | 0 | 0.2 | 0.5 | 1 | 0.092 |

$\widehat{\mathsf{\mu}}$ | 139.413 | 132.468 | 128.227 | 122.340 | 135.257 |

$\widehat{\mathsf{\sigma}}$ | 65.360 | 56.354 | 55.286 | 52.815 | 59.122 |

$\mathsf{\alpha}$ | 0 | 0.2 | 0.5 | 1 |

$\widehat{\mathsf{\mu}}$ | 126.380 | 126.453 | 126.705 | 126.461 |

$\widehat{\mathsf{\sigma}}$ | 49.747 | 51.759 | 54.615 | 56.341 |

$\widehat{\mathsf{\xi}}$ | 0.404 | 0.438 | 0.497 | 0.506 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Seo, Y.; Hwang, J.; Kim, B.
Extreme Precipitation Frequency Analysis Using a Minimum Density Power Divergence Estimator. *Water* **2017**, *9*, 81.
https://doi.org/10.3390/w9020081

**AMA Style**

Seo Y, Hwang J, Kim B.
Extreme Precipitation Frequency Analysis Using a Minimum Density Power Divergence Estimator. *Water*. 2017; 9(2):81.
https://doi.org/10.3390/w9020081

**Chicago/Turabian Style**

Seo, Yongwon, Junshik Hwang, and Byungsoo Kim.
2017. "Extreme Precipitation Frequency Analysis Using a Minimum Density Power Divergence Estimator" *Water* 9, no. 2: 81.
https://doi.org/10.3390/w9020081