# On Complex Network Construction of Rain Gauge Stations Considering Nonlinearity of Observed Daily Rainfall Data

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Theory

#### 2.1. BDS Statistic and Nonlinearity Test

- M = N (m − 1): The number of state vector points in m-dimensional (m = embedding dimension).
- r: Radius for determining the number of state vectors points.
- $||\xb7||$: the sup-norm.

#### 2.2. Pearson Correlation and Mutual Information

#### 2.3. Graph Theory and Complex Network

#### 2.3.1. General

#### 2.3.2. Centrality $\left({\mathrm{D}}_{\mathrm{c}}\right)$

## 3. Application and Results

#### 3.1. Study Area and Data

#### 3.2. Nonlinearity of Rainfall

#### 3.3. Analysis and Results

#### 3.4. Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Burgueno, A.; Vilar, E.; Puigcerver, M. Spectral Analysis 49 Years of Rainfall Rate and Relation to Fade Dynamics. IEE Trans. Commun.
**1990**, 9, 1359–1366. [Google Scholar] [CrossRef] - Goyal, M.K. Monthly rainfall prediction using wavelet regression and neural network: An analysis of 1901–2002 data, Assam, India. Theor. Appl. Climatol.
**2014**, 118, 25–34. [Google Scholar] [CrossRef] - Kyoung, M.S.; Kim, H.S.; Sivakumar, B.; Singh, V.P.; Ahn, K.S. Dynamic characteristics of monthly rainfall in the Korean Penisula under climate change. Stoch. Environ. Res. Risk Assess.
**2011**, 25, 613–625. [Google Scholar] [CrossRef] - Olayide, O.E.; Alabi, T. Between rainfall and food poverty: Assessing vulnerability to climate change in an agricultural economy. J. Clean. Prod.
**2018**, 198, 1–10. [Google Scholar] [CrossRef] - DehghanSh, K.S.; Eslamian, S.; Gandomkar, A.; Marani-Barzani, M.; Amoushahi-Khouzani, M.; Singh, V.P.; Ostad-Ali-Askari, K. Change in Temperature and Precipitation with the Anaylsis of Geomorphic Basin Chaos in Shiranz. Iran Int. J. Constr. Res. Civ. Eng. (IJCRCE)
**2017**, 3, 50–57. [Google Scholar] [CrossRef] - Krajewski, W.F.; Ciach, G.J.; Habib, E. An analysis of small-scale rainfall variability in different climatic regimes. Hydrol. Sci. J.
**2002**, 48, 151–162. [Google Scholar] [CrossRef] - Di Piazza, A.; Conti, F.L.; Noto, L.V.; Viola, F.; La Loggia, G. Comparative analysis of different techiques for spatial interpolation of rainfall data to creat a serially complete monthly time series of precipitation for Sicily, Italy. Int. J. Appl. Earth Obs. Geoinf.
**2011**, 13, 396–408. [Google Scholar] [CrossRef] - Tokar, A.S.; Markus, M. Precipiation-Runoff Modelling using Artificial Neural Networks and Conceptual Models. J. Hydrol. Eng.
**2000**, 5, 156–161. [Google Scholar] [CrossRef] - Duffourg, F.; Ducrocq, V. Assessment of the water supply to Mediterranean heavy precipitation: A method based on finely designed water budgets. Atmos. Sci. Lett.
**2013**, 14, 133–138. [Google Scholar] [CrossRef] - Binti Sa’adin, S.L.; Kaewunruen, S.; Jaroszweski, D. Heavy rainfall and flood vulnerability of Singapore-Malaysia high speed rail system. Aust. J. Civ. Eng.
**2016**, 14, 123–131. [Google Scholar] [CrossRef] - David, C.C.; Dotson, H.W. Rain Gage Network Size for Automated Flood Warning System, Conference Proceeding of Engineering Hydrology; ASCE: Reston, VA, USA, 1993. [Google Scholar]
- Ministry of Land, Infrastructure and Transport (MLIT). Han River Watershed Research Hydraulic and Hydrological Research Report; MLIT: Tokyo, Japan, 2004.
- Dyck, G.E.; Gray, D.M. Spatial Characteristics of Prairie Rainfall; American Meteorological Society: Tronoto, ON, Canada, 1997; pp. 25–27. [Google Scholar]
- Brock, W.A.; Heish, D.A.; Lebaron, B. Nonlinear Dynamics Chaos and Instability Statistical Theory and Economic Evidence; The MIT Press Publisher: Cambridge, MA, USA, 1991. [Google Scholar]
- Jha, S.K.; Zhao, H.; Woldemeskel, F.M.; Sivakumar, B. Network Theory and spatial rainfall connections: An Interpretation. J. Hydrol.
**2015**, 527, 13–19. [Google Scholar] [CrossRef] - Luk, K.C.; Ball, J.E.; Sharma, A. A Study of optimal model lag and spatial inputs to artificial neural network for rainfall forecasting. J. Hydrol.
**2000**, 227, 56–65. [Google Scholar] [CrossRef] - Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.U. Complex networks:Structure and dynamics. Phys. Rep.
**2006**, 424, 175–308. [Google Scholar] [CrossRef] - Latora, V.; Nicosia, V.; Russo, G. Complex Networks Principles, Methods and Applications; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Schweitzer, F.; Fagiolo, G.; Sornette, D.; Vega-Redondo, F.; Vespignani, A.; White, D.R. Economic Networks: The New Challenges. Science
**2009**, 325, 422–425. [Google Scholar] [CrossRef] [PubMed] - Pagani, G.A.; Aiello, M. The Power Grid as a complex Netowrk: A survey. Physica A: Stat. Mech. Appl.
**2013**, 392, 2688–2700. [Google Scholar] [CrossRef] - Mason, O.; Verwoerd, M. Graph theory and networks in Biology. IET Syst. Biol.
**2007**, 1, 89–119. [Google Scholar] [CrossRef] [Green Version] - Milo, R.; Shen-Orr, S.; Itzkovitz, S.; Kashtan, N.; Chklovskii, D.; Alon, U. Network Motifs: Simple Building Block of Complex Networks. Science
**2002**, 298, 824–827. [Google Scholar] [CrossRef] - Yazdani, A.; Jeffrey, P. Complex network analysis of water distributuion systems. Chaos
**2011**, 21, 01611. [Google Scholar] [CrossRef] [PubMed] - Boers, N.; Bookhagen, B.; Marwan, N.; Kurths, J.; Marengo, J. Complex networks identiy spatial patterns of extreme rainfall events of the South American Monsoon System. Geophys. Reasearch Lett.
**2013**, 40, 4386–4392. [Google Scholar] [CrossRef] - Sivakumar, B.; Woldemeskel, F.M. Complex networks for streamflow dynamics. Hydrol. Earth Syst. Sci.
**2014**, 11, 4565–4578. [Google Scholar] [CrossRef] - Halverson, J.M.; Fleming, S.W. Complex network theory, streamflow, and hydrometric monitoring system design. Hydrol. Earth Syst. Sci.
**2015**, 19, 3301–3318. [Google Scholar] [CrossRef] [Green Version] - Cover, T.M.; Tomas, J.A. Elements of Information Theory, 2nd ed.; Schilling, D.L., Ed.; Wiley Series in Telecommunications Press: New York, NY, USA, 2006. [Google Scholar]
- Numata, J.; Ebenhöh, O.; Knapp, E.W. Measuring correlation in Metabolomic networks with Mutual Information. Genome Inform. Ser.
**2008**, 20, 112–122. [Google Scholar] [CrossRef] - Dadgostar, M.; Einalou, Z.; Setarehdam, S.K.; Keskin-Ergen, H.Y.; Akin, A. Comparison of Mutual Information and Partial Correlation for Functional Connectivity in fNIRS. In Proceedings of the 21th Iranian Conference on Electric Engineering, Mashhad, Iran, 14–16 May 2013. [Google Scholar]
- Donges, J.F.; Zou, Y.; Marwan, N.; Kurths, J. Complex networks in climate dynamics Comparing linear and nonlinear network construction methods. Eur. Phys. J. Spec. Top.
**2009**, 174, 157–179. [Google Scholar] [CrossRef] - Wang, J.; He, J.M. Correlation and Interdependence Structure in Stock Market: Based on Information Theory and Complex Networks. In Proceedings of the 17th International Conference on Control, Automation and Systmes (ICCAS 2017), Jeju, Korea, 18–21 October 2017. [Google Scholar]
- Zhang, W.; Ma, J.; Ideker, T. Classifying tumors by supervised networks propagation. Bioinformatics
**2018**, 34, 484–493. [Google Scholar] [CrossRef] [PubMed] - Kroll, M.H.; Emancipator, K. A Theoretical Evaluation of Linearity. Clin. Chem.
**1993**, 39, 405–413. [Google Scholar] [PubMed] - Strogatz, S.H. Nonlinear Dynamics and Chaos with Application to Physics, Biology, Chemistry and Engineering; Westview Press: Philadelphia, PA, USA, 2015. [Google Scholar]
- Brock, W.A.; Scheinkman, J.A.; Dechert, W.D.; LeBaron, B. A test for independence based on the correlation dimension. Econom. Rev.
**1996**, 15, 197–235. [Google Scholar] [CrossRef] - Kim, H.S.; Eykholt, R.; Salas, J.D. Delay time window and plateau onset of the correlation dimension for small data sets. Phys. Rev. E
**1998**, 58, 5676–5682. [Google Scholar] [CrossRef] - Kim, H.S.; Eykholt, R.; Salas, J.D. Nonliear dynamics, delay times and embedding windows. Phys. D
**1999**, 127, 48–60. [Google Scholar] [CrossRef] - Kim, S.; Noh, H.; Kang, N.; Lee, K.; Kim, Y.; Lim, S.; Lee, D.R.; Kim, H.S. Noise Reduction Analysis of Radar Rainfall Using Chaotic Dynamics and Filtering Techniques; Hindawi Publishing Corporation: London, UK, 2014; pp. 1–10. [Google Scholar]
- Kim, H.S.; Kang, D.S.; Kim, J.H. The BDS statistic Application to Hydrologic Data. J. Korea Water Resour. Assoc.
**2003**, 31, 769–777. [Google Scholar] - Kim, K.H.; Han, D.G.; Kim, J.W.; Lim, J.H.; Lee, J.S.; Kim, H.S. Modelling and Residual Analysis for Water Level Series of Upo Wetland. J. Wetl. Res.
**2018**, 21, 66–76. [Google Scholar] [CrossRef] - Bavelas, A. A mathematical model for group structure. Hum. Org.
**1948**, 7, 16–30. [Google Scholar] [CrossRef] - Leavitt, H.J. Some effects of certain communication patterns on group performance. J. Abnorm. Soc. Psych.
**1951**, 46, 38–50. [Google Scholar] [CrossRef] - Jeong, H.; Tomber, B.; Albert, R.; Oltavi, Z.N.; Barabási, A.L. The large-scale organization of metabolic networks. Nature
**2000**, 40, 651–654. [Google Scholar] [CrossRef] [PubMed] - Newman, M.E.J. The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA
**2001**, 98, 404–409. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**The 55 rainfall gauge stations in the study area (Latitude: 34.3959–38.2509° N, Longitude: 126.3812–129.4128° E).

**Figure 3.**Mutual information—Pearson correlation graph (09: Geoje): The X-axis is the Pearson coefficient and the Y-axis is the mutual information. In the graph, the two axes have different ranges (X: 0.0–1.0, Y: 0.0–1.5).

**Figure 4.**The number of links according to threshold: (

**a**) mutual information; (

**b**) Pearson correlation.

**Figure 5.**Selection of links according to threshold: the mutual information and Pearson coefficient between stations are calculated as links. According to the threshold, the values, which is bigger than threshold, are filled with red color and the others remain as white color.

**Figure 6.**Complex network connected by threshold 0.7: (

**a**) mutual information; (

**b**) Pearson correlation.

**Figure 7.**Estimation of centrality and rank of station by Pearson correlation: The X-axis mean the rank of station and the Y-axis is the values of centrality. The number upon the bar mean the stations which belong to the rank.

**Figure 8.**Estimation of centrality and rank of station by mutual information: on the X-axis is the rank of station and on the Y-axis are the values of centrality. The number upon the bar mean the stations which belong to the rank.

**Figure 9.**Locations of the most important station according to the threshold. The stations that have the highest value of centrality are expressed in the map according to the threshold (0.3, 0.4, 0.5, 0.6, 0.7). The location of the station in the case of mutual information is in the central of the Korean peninsula. The result of Pearson correlation shows that locations of the highest ranked station are moving into the south part of the Korean peninsula.

**Table 1.**Basic statistics of daily rainfall series of 55 rainfall gaging stations: all basic statistics of each station are in Supplementary Materials, Appendix A.

Statistics | Max | Mean | Standard Deviation | Coefficient of Variation |
---|---|---|---|---|

Value (Range) | 122.40–870.50 | 0.35–5.11 | 3.54–18.54 | 3.31–10.00 |

Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Station | Sokcho | Wonju | Inje | Chun cheon | Hong cheon | Suwon | Yan pyeong | Icheon | Geoje | Geo chang |

Number | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Station | Namhae | Miryang | San cheong | Jinju | Tong yeong | Hap cheon | Gumi | Mun gyeong | Yeong deok | Yeongju |

Number | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

Station | Yeong cheon | Uljin | Uiseong | Pohang | Goheung | Mokpo | Yeosu | Wando | Jang heung | Juam |

Number | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

Station | Haenam | Gunsan | Namwon | Buan | Imsil | Jeonju | Jeong eup | Geumsan | Bor yeong | Buyeo |

Number | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

Station | Seosan | Cheonan | Boeun | Jecheon | Cheong ju | Chupung yeong | Chungju | Ganghwa | Incheon | Gwangju |

Number | 51 | 52 | 53 | 54 | 55 | |||||

Station | Daegu | Daejeon | Busan | Seoul | Ulsan |

**Table 3.**Brock–Dechert–Scheinkman (BDS) statistic results of observed daily rainfall (09: Geoje): all values of BDS statistics results are out of Confidence Interval. The null hypothesis is rejected, and observation data is determined as nonlinear data. The results of the other stations are shown in the Supplementary Materials, Appendix B.

Index | $\mathbf{r}=0.5\mathbf{s}$ | $\mathbf{r}=1.0\mathbf{s}$ | $\mathbf{r}=1.5\mathbf{s}$ | $\mathbf{r}=2.0\mathbf{s}$ | C.I |
---|---|---|---|---|---|

$m=2$ | 22.978 | 21.580 | 20.429 | 20.406 | (−1.96, 1.96) |

$m=3$ | 18.091 | 17.193 | 16.335 | 16.254 | (−1.96, 1.96) |

$m=4$ | 15.559 | 14.115 | 13.364 | 13.318 | (−1.96, 1.96) |

$m=5$ | 14.740 | 13.520 | 13.071 | 12.956 | (−1.96, 1.96) |

**Table 4.**The first station of centrality and links. The most important stations and their links are expressed in the map according to the threshold (0.4 to 0.7). In the case of threshold 0.3, many stations are selected and each of the chosen stations connected with all stations in both cases (mutual information and Pearson coefficient).

Threshold | Mutual Information | Pearson Correlation |
---|---|---|

0.4 | ||

0.5 | ||

0.6 | ||

0.7 |

**Table 5.**The most important station according to threshold. The stations which have the highest value of centrality are chosen according to the threshold (0.3, 0.4, 0.5, 0.6, 0.7). The mutual information results have consistent results, but the Pearson correlation results have variability.

Method | Mutual Information | Pearson Correlation | |
---|---|---|---|

Threshold | 0.3 | # 10, # 17, # 18, # 20, # 21, # 23, # 32, # 33, # 34, # 35, # 36, # 38, # 43, # 44, # 45, # 46, # 47, # 52 | # 18, # 20, # 32, # 38, # 40, # 43, # 45, # 52 |

0.4 | # 18 | # 18, # 20 | |

0.5 | # 18 | # 17 | |

0.6 | # 18 | # 10 | |

0.7 | # 18 | # 10, # 14 |

© 2019 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**

Kim, K.; Joo, H.; Han, D.; Kim, S.; Lee, T.; Kim, H.S.
On Complex Network Construction of Rain Gauge Stations Considering Nonlinearity of Observed Daily Rainfall Data. *Water* **2019**, *11*, 1578.
https://doi.org/10.3390/w11081578

**AMA Style**

Kim K, Joo H, Han D, Kim S, Lee T, Kim HS.
On Complex Network Construction of Rain Gauge Stations Considering Nonlinearity of Observed Daily Rainfall Data. *Water*. 2019; 11(8):1578.
https://doi.org/10.3390/w11081578

**Chicago/Turabian Style**

Kim, Kyunghun, Hongjun Joo, Daegun Han, Soojun Kim, Taewoo Lee, and Hung Soo Kim.
2019. "On Complex Network Construction of Rain Gauge Stations Considering Nonlinearity of Observed Daily Rainfall Data" *Water* 11, no. 8: 1578.
https://doi.org/10.3390/w11081578