# Spatial Disaggregation of Areal Rainfall Using Two Different Artificial Neural Networks Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Artificial Neural Networks

#### 2.1. Multilayer Perceptron (MLP) Model

#### 2.2. Kohonen Self-Organizing Feature Map (KSOFM) Model

_{1}-by-n

_{1}] matrices. The KSOFM model is a simple yet powerful learning process and an effective clustering method, and uses a neighborhood function to preserve the topological properties of the input space. It can transform high dimensional input patterns into the responses of two-dimensional arrays of neurons and perform this transformation adaptively in a topologically ordered fashion based on similarity. Detailed information on the KSOFM model can be found in Kohonen [55,56], Principe et al. [57], and Hsu et al. [58].

## 3. Case Study

^{2}in area, represents 77.1% of the total area, 612.86 km

^{2}, of Kunwi-gun county. The Wi-stream catchment is narrow from south to north and long from east to west. The central part of the Wi-stream catchment is quite flat and suffers from storm and flood damages every year. There are six river stage stations, six groundwater stations, 11 rainfall stations, and 11 evaporation stations in the Wi-stream catchment. The stream network consists of one main stream and one tributary [59]. The hydrological data of the Wi-stream catchment, such as rainfall, river stage, discharge, and groundwater table, have been recorded since 1982.

_{mean}, X

_{max}, X

_{min}, S

_{x}, C

_{v}, C

_{sx}, and SE denote, respectively, the mean, maximum, minimum, standard deviation, coefficient of variation, skewness coefficient and standard error values of training, cross-validation, and testing data. The estimated values were compared with observed values using four different performance evaluation criteria: the Nash-Sutcliffe efficiency [62] (NS), root mean square error (RMSE), mean absolute error (MAE), and average performance error (APE). As a measure of the accuracy of any hydrologic model, NS is one of the most widely used criteria for calibration and evaluation of hydrological models [63]. It has been shown that NS alone cannot define which model is better than others. The various evaluation criteria (e.g., RMSE, MAE, and APE) must be used to define the model performance. The NS, RMSE, MAE, and APE evaluation criteria quantify the efficiency of a model in capturing extremely complex, dynamic, nonlinear, and fragmented relationships. A model, which is efficient in capturing the complex relationship among the various input and output variables involved in a particular problem, must be considered [64]. Table 2 shows mathematical expressions of performance evaluation criteria.

Division | Number of Data | Statistical Indices of Areal Rainfall | ||||||
---|---|---|---|---|---|---|---|---|

X_{mean} | X_{max} | X_{min} | S_{x} | C_{v} | C_{sx} | SE | ||

Training | 338 | 3.26 | 27.76 | 0.00 | 4.21 | 1.12 | 2.32 | 0.21 |

Cross-validation | 77 | 2.10 | 16.68 | 0.00 | 3.12 | 1.32 | 2.62 | 0.34 |

Testing | 91 | 3.62 | 19.56 | 0.00 | 4.13 | 1.13 | 1.46 | 0.42 |

Evaluation Criteria | Equation |
---|---|

NS | $\text{1}-\frac{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{{\text{[y}}_{\text{i}}\text{(x)}-{\widehat{\mathrm{y}}}_{i}\text{(x)]}}^{\text{2}}}}{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{{\text{[y}}_{\text{i}}\text{(x)}-{\text{u}}_{\text{y}}\text{]}}^{\text{2}}}}$ |

RMSE | $\sqrt{\frac{\text{1}}{\text{n}}{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{{\text{[y}}_{\text{i}}\text{(x)}-{\widehat{\mathrm{y}}}_{\mathrm{i}}\text{(x)]}}^{\text{2}}}}$ |

MAE | $\frac{\text{1}}{\text{n}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{y}}_{\mathrm{i}}(x)-{\widehat{\mathrm{y}}}_{\mathrm{i}}(\mathrm{x})\right|}$ |

APE | $\frac{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}\left|{\text{y}}_{\text{i}}(\mathrm{x})-{\widehat{\mathrm{y}}}_{\mathrm{i}}(\mathrm{x})\right|}}{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{y}}_{\text{i}}\text{(x)}}}\times 100$ |

_{i}(x) = the observed hourly rainfall (mm); y

_{i}(x) = the estimated hourly rainfall (mm); u

_{v}= the mean of observed hourly rainfall (mm); and n = the total number of hourly rainfall values considered.

## 4. Applications and Results

#### 4.1. Selection of Optimal MLP Models for Estimating Areal Rainfall

_{kj}= the connection weights between the hidden and output layers; W

_{ji}= the connection weights between the input and hidden layers; $\text{X(t)}$ = the time series data of input variables; ${\text{B}}_{\text{1}}$ = the bias in hidden layer; and ${\text{B}}_{\text{2}}$ = the bias in output layer. Figure 3 shows the structure of MLP (11-5-1) developed for estimating areal rainfall in this study.

**Figure 2.**Influence of the number of hidden nodes for three training algorithms (test period). (

**a**) NS; (

**b**) MAF; (

**c**) RMSE; (

**d**) APE.

_{α}) was computed for the specific level of significance. If the computed value of z statistic is greater than the critical value of z statistic (z

_{α}), the null hypothesis, that the two independent samples are from the same population, should be rejected and the alternative hypothesis should be accepted.

_{α}), z

_{0.05}= 1.960, was computed for the five percent level of significance. Since the computed values of z statistic for both stations were not significant, the null hypothesis was accepted for areal rainfall using the MLP model.

Model | Networks | Training Algorithms | Level of Significance | Mann-Whitney U test | ||
---|---|---|---|---|---|---|

Critical z Statistic | Computed z Statistic | Null Hypothesis | ||||

MLP | 11-5-1 | Conjugate gradient | 0.05 | 1.960 | −0.287 | Accept |

11-3-1 | Levenberg–Marquardt | 0.05 | 1.960 | −0.617 | Accept | |

11-5-1 | Quickprop | 0.05 | 1.960 | −0.515 | Accept |

#### 4.2. Evaluation for Spatial Disaggregation of Areal Rainfall Using MLP Model

**Figure 5.**Influence of individual rainfall stations for three training algorithms of MLP (test period). (

**a**) NS; (

**b**) MAF; (

**c**) RMSE; (

**d**) APE.

#### 4.3. Evaluation for Spatial Disaggregation of Areal Rainfall Using KSOFM Model

_{kj}= the connection weights between the Kohonen and hidden layers; S

_{j}= the results calculated from the Euclidean distance (d

_{j}) and the Kohonen layer; ${\text{\Phi}}_{1}(\cdot )$ = the linear sigmoid transfer function of hidden layer; ${\text{\Phi}}_{2}(\cdot )$ = the linear sigmoid transfer function of output layer; B

_{1}= the bias in hidden layer; B

_{2}= the bias in output layer; and W

_{lk}= the connection weights between the hidden and output layers. The Euclidean distance between the input and Kohonen nodes can be written as:

**Figure 6.**Structure of KSOFM (1-[5 X 5]-5-11) developed for spatial disaggregation of areal rainfall.

**Figure 7.**Influence of individual rainfall stations for three training algorithms of KSOFM1 (test period). (

**a**) NS; (

**b**) MAF; (

**c**) RMSE; (

**d**) APE.

**Figure 8.**Influence of individual rainfall stations for three training algorithms of KSOFM2 (test period). (

**a**) NS; (

**b**) MAF; (

**c**) RMSE; (

**d**) APE.

**Figure 9.**Rainfall box plots for Euiheung (No.8) station (test period). (

**a**) Conjugate gradient; (

**b**) Levenberg–Marquardt; (

**c**) Quickprop.

**Figure 10.**Rainfall box plots for Hwasu (No.9) station (test period). (

**a**) Conjugate gradient; (

**b**) Levenberg–Marquardt; (

**c**) Quickprop.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

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Kim, S.; Singh, V.P.
Spatial Disaggregation of Areal Rainfall Using Two Different Artificial Neural Networks Models. *Water* **2015**, *7*, 2707-2727.
https://doi.org/10.3390/w7062707

**AMA Style**

Kim S, Singh VP.
Spatial Disaggregation of Areal Rainfall Using Two Different Artificial Neural Networks Models. *Water*. 2015; 7(6):2707-2727.
https://doi.org/10.3390/w7062707

**Chicago/Turabian Style**

Kim, Sungwon, and Vijay P. Singh.
2015. "Spatial Disaggregation of Areal Rainfall Using Two Different Artificial Neural Networks Models" *Water* 7, no. 6: 2707-2727.
https://doi.org/10.3390/w7062707