# Daily Runoff Forecasting Model Based on ANN and Data Preprocessing Techniques

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data-Driven Models

#### 2.1. NLPM-ANN Model

_{d}) that is transformed to the series of the seasonal expectations of the output (q

_{d}) through an undefined relation. The second part, which is the input perturbations (P

_{i}-p

_{d}), is transformed into the output perturbations (Q

_{i}-q

_{d}) through ANN. The total output is the sum of the seasonal expectations of the output and the output perturbations.

#### 2.2. Singular Spectrum Analysis

_{0}, f

_{1}, …, f

_{N}

_{−1}} (f

_{i}≠ 0), the length of series is N (>2). Given a window length L, the one-dimensional time series can be transferred into a sequence of L-dimensional vectors

**X**

_{i}= {f

_{i}

_{−1}, …, f

_{i}

_{+L−2}}

^{T}, (I = 1, …, K = N−L+1). The K vectors

**X**

_{i}will form the columns of the (L × K) trajectory matrix:

**X**is conducted. Let

**S**=

**XX**

^{T}. The eigenvalues and eigenvectors of

**S**can be calculated, and these eigenvalues range in the decreasing order of magnitude. According to the conventional computation of EOF, an expansion of the matrix

**X**is represented as:

_{k}, thus the initial series F is decomposed into the sum of L series:

#### 2.3. Artificial Neural Network

_{t}is the input data; T is the length of lead time; φ denotes transfer functions; w

_{ji}are the weights defining the link between the ith node of the input layer and the jth of the hidden layer; θ

_{j}are biases associated with the jth node of the hidden layer; ${w}_{j}^{out}$ are the weights associated with the connection between the jth node of the hidden layer and the node of the output layer; and θ

_{0}is the bias at the output node. The Levenberg–Marquardt algorithm is chosen to adjust the values of w and θ in this study [32].

#### 2.4. Proposed SSA-ANN Models

#### 2.5. Evaluation of Model Performances

- (1)
- Determination coefficient (or Nash-Sutcliffe criterion) (R
^{2})$${R}^{2}=(1-\frac{{\displaystyle {\sum}_{t=1}^{n}{({Q}_{t}-{Q}_{t}^{\prime})}^{2}}}{{\displaystyle {\sum}_{t=1}^{n}{({Q}_{t}-\overline{{Q}_{t}})}^{2}}})$$ - (2)
- Water balance coefficient (WB)$$WB=\frac{{\displaystyle {\sum}_{t=1}^{n}{Q}_{t}^{\prime}}}{{\displaystyle {\sum}_{t=1}^{n}{Q}_{t}}}$$

^{2}and WB are closer to one, the better the prediction results that are obtained.

## 3. Comparative Study

#### 3.1. Data

_{x}), maximum (X

_{max}), and minimum (X

_{min}). As shown in Table 1, the training data does not cover the cross-validation or testing data totally. In order to ensure the extrapolation ability of ANN and avoid numerical difficulties during calculation, all data are scaled to the interval [−0.9, 0.9] by normalization.

Watershed and Datasets | Statistical Parameters | Data Period | |||||
---|---|---|---|---|---|---|---|

μ | S_{x} | X_{max} | X_{min} | ||||

Jiahe area: 5578 km^{2} | rainfall (mm) | whole data | 2.3 | 5.9 | 71.4 | 0 | January 1980–December 1990 |

training data | 2.3 | 6.0 | 68.9 | 0 | |||

cross-validation data | 2.3 | 6.2 | 71.4 | 0 | |||

testing data | 2.1 | 5.3 | 44.2 | 0 | |||

runoff (m^{3}) | whole data | 58.7 | 125.1 | 2620 | 6.5 | ||

training data | 61.9 | 141.6 | 2620 | 6.5 | |||

cross-validation data | 55.3 | 99.6 | 1220 | 7.9 | |||

testing data | 50.7 | 76.4 | 1080 | 10.1 | |||

Laoguanhe area: 4217 km^{2} | rainfall (mm) | whole data | 2.2 | 6.4 | 69.4 | 0 | January 1980–December 1990 |

training data | 2.3 | 6.8 | 69.2 | 0 | |||

cross-validation data | 2.0 | 5.8 | 56.0 | 0 | |||

testing data | 2.0 | 5.7 | 69.4 | 0 | |||

runoff (m^{3}) | whole data | 27.1 | 73.6 | 1460 | 0.1 | ||

training data | 33.5 | 84.1 | 1460 | 0.4 | |||

cross-validation data | 16.8 | 50.6 | 586 | 0.1 | |||

testing data | 14.8 | 46.1 | 793 | 0.2 | |||

Baohe area: 3415 km^{2} | rainfall (mm) | whole data | 2.5 | 6.9 | 80.6 | 0 | January 1980–December 1990 |

training data | 2.5 | 7.1 | 80.6 | 0 | |||

cross-validation data | 2.2 | 6.0 | 51.3 | 0 | |||

testing data | 2.6 | 6.8 | 80.5 | 0 | |||

runoff (m^{3}) | whole data | 46.5 | 129.4 | 4020 | 0 | ||

training data | 49.7 | 150.7 | 4020 | 1.2 | |||

cross-validation data | 31.4 | 54.8 | 523 | 3.8 | |||

testing data | 50.3 | 96.8 | 2010 | 0.0 | |||

Mumahe area: 1224 km^{2} | rainfall (mm) | whole data | 3.2 | 8.8 | 132.8 | 0 | January 1980–December 1990 |

training data | 3.2 | 8.6 | 132.8 | 0 | |||

cross-validation data | 3.3 | 9.3 | 98.6 | 0 | |||

testing data | 2.9 | 9.1 | 94.4 | 0 | |||

runoff (m^{3}) | whole data | 39.3 | 80.3 | 1270 | 1.2 | ||

training data | 41.0 | 80.8 | 1270 | 1.2 | |||

cross-validation data | 40.6 | 82.1 | 796 | 4.6 | |||

testing data | 32.1 | 76.4 | 990 | 2 | |||

Nianyushan area: 924 km^{2} | rainfall (mm) | whole data | 3.8 | 11.6 | 269.5 | 0 | January 1975–December 1999 |

training data | 3.9 | 12.2 | 269.5 | 0 | |||

cross-validation data | 3.3 | 9.3 | 102.5 | 0 | |||

testing data | 3.7 | 10.8 | 144.7 | 0 | |||

runoff (m^{3}) | whole data | 18.5 | 62.1 | 2095 | 0 | ||

training data | 19.8 | 68.3 | 2095 | 0 | |||

cross-validation data | 13.5 | 33.2 | 508 | 0 | |||

testing data | 17.6 | 55.9 | 822 | 0 | |||

Gaoguan area: 303 km^{2} | rainfall (mm) | whole data | 4.2 | 12.5 | 179.1 | 0 | January 1984–December 1999 |

training data | 4.4 | 12.8 | 179.1 | 0 | |||

cross-validation data | 3.5 | 11.3 | 143.8 | 0 | |||

testing data | 4.2 | 12.7 | 116.0 | 0 | |||

runoff (m^{3}) | whole data | 5.8 | 15.1 | 246 | 0 | ||

training data | 5.7 | 14.2 | 237 | 0 | |||

cross-validation data | 5.1 | 13.5 | 246 | 0 | |||

testing data | 7.7 | 20.5 | 214 | 0 | |||

Shimen area: 271.25 km^{2} | rainfall (mm) | whole data | 3.8 | 11.4 | 141.3 | 0 | January 1989–December 1999 |

training data | 3.5 | 10.1 | 114.9 | 0 | |||

cross-validation data | 5.1 | 15.1 | 141.3 | 0 | |||

testing data | 3.8 | 11.8 | 116.8 | 0 | |||

runoff (m^{3}) | whole data | 4.9 | 15.2 | 296 | 0 | ||

training data | 3.7 | 9.9 | 150 | 0 | |||

cross-validation data | 8.7 | 25.1 | 296 | 0 | |||

testing data | 5.5 | 17.9 | 172 | 0 | |||

Tiantang area: 220 km^{2} | rainfall (mm) | whole data | 3.7 | 12.1 | 193.4 | 0 | January 1973–December 1984 |

training data | 3.6 | 11.6 | 175.0 | 0 | |||

cross-validation data | 3.7 | 11.4 | 151.7 | 0 | |||

testing data | 4.2 | 14.7 | 193.4 | 0 | |||

runoff (m^{3}) | whole data | 6.1 | 18.4 | 535 | 0 | ||

training data | 5.6 | 16.5 | 400 | 0 | |||

cross-validation data | 5.6 | 16.5 | 378 | 0.3 | |||

testing data | 8.2 | 25.6 | 535 | 0.3 |

#### 3.2. Determination of Model Inputs

#### 3.3. Data Preprocessing

**Table 2.**Cross-correlation function (CCF) values between each decomposed component and original series.

Watershed | Decomposed Components | L | p | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

Jiahe | rainfall | −0.26 | −0.27 | −0.19 | −0.05 | 0.13 | 0.36 | 0.50 | 0.55 | – | – | 8 | 4 |

runoff | −0.14 | −0.15 | −0.11 | −0.05 | 0.05 | 0.18 | 0.39 | 0.55 | 0.77 | – | 9 | 5 | |

Laoguanhe | rainfall | −0.36 | −0.33 | −0.24 | −0.06 | 0.12 | 0.33 | 0.47 | 0.53 | – | – | 8 | 4 |

runoff | −0.15 | −0.15 | −0.10 | 0.00 | 0.14 | 0.35 | 0.55 | 0.77 | – | – | 8 | 4 | |

Baohe | rainfall | −0.26 | −0.26 | −0.18 | −0.04 | 0.14 | 0.35 | 0.50 | 0.60 | – | – | 8 | 4 |

runoff | −0.18 | −0.20 | −0.16 | −0.08 | 0.04 | 0.16 | 0.33 | 0.54 | 0.76 | – | 9 | 5 | |

Mumahe | rainfall | −0.34 | −0.32 | −0.22 | −0.06 | 0.13 | 0.34 | 0.47 | 0.52 | – | – | 8 | 4 |

runoff | −0.15 | −0.18 | −0.14 | −0.09 | −0.01 | 0.11 | 0.25 | 0.41 | 0.56 | 0.71 | 10 | 5 | |

Nianyushan | rainfall | −0.33 | −0.33 | −0.26 | −0.13 | 0.02 | 0.19 | 0.35 | 0.47 | 0.51 | – | 9 | 5 |

runoff | −0.22 | −0.22 | −0.16 | −0.03 | 0.15 | 0.34 | 0.54 | 0.68 | – | – | 8 | 4 | |

Gaoguan | rainfall | −0.32 | −0.37 | −0.30 | −0.18 | −0.07 | 0.09 | 0.23 | 0.37 | 0.46 | 0.43 | 10 | 5 |

runoff | −0.14 | −0.19 | −0.17 | −0.12 | −0.03 | 0.09 | 0.23 | 0.42 | 0.58 | 0.67 | 10 | 5 | |

Shimen | rainfall | −0.34 | −0.34 | −0.32 | −0.28 | 0.01 | 0.19 | 0.35 | 0.47 | 0.48 | – | 9 | 5 |

runoff | −0.21 | −0.23 | −0.18 | −0.09 | 0.04 | 0.19 | 0.39 | 0.58 | 0.66 | – | 9 | 5 | |

Tiantang | rainfall | −0.32 | −0.34 | −0.19 | 0.03 | 0.28 | 0.46 | 0.53 | – | – | – | 7 | 4 |

runoff | −0.31 | −0.31 | −0.16 | 0.03 | 0.25 | 0.46 | 0.62 | – | – | – | 7 | 4 |

## 4. Results Analysis

^{2}and WB of eight watersheds are 70.16% and 0.879 by ANN, and are increased to 75.86% and 1.155 by NLPM-ANN, and 80.62% and 1.04 by SSA-ANN1, respectively. In the Tiantang watershed, the performance of the NLPM-ANN and SSA-ANN1 models is improved significantly, so the R

^{2}value increased from 59.79% to 81.96% and 79.54%, respectively, during the testing period.

Watershed | ANN | NLPM-ANN | SSA-ANN1 | SSA-ANN2 | |||||
---|---|---|---|---|---|---|---|---|---|

R^{2} (%) | WB | R^{2} (%) | WB | R^{2} (%) | WB | R^{2} (%) | WB | ||

Jiahe | calibration | 68.19 | 1.023 | 85.46 | 1.015` | 80.97 | 0.982 | 96.09 | 1.013 |

testing | 61.48 | 0.866 | 61.31 | 1.119 | 74.91 | 0.975 | 92.40 | 1.013 | |

Laoguanhe | calibration | 69.72 | 1.048 | 85.66 | 1.042 | 82.29 | 0.972 | 96.31 | 1.186 |

testing | 60.42 | 1.058 | 68.25 | 1.412 | 78.44 | 1.464 | 93.20 | 1.407 | |

Baohe | calibration | 64.75 | 0.975 | 70.93 | 1.039 | 88.50 | 1.029 | 94.01 | 1.006 |

testing | 68.62 | 0.667 | 69.38 | 0.893 | 74.03 | 0.927 | 94.31 | 0.956 | |

Mumahe | calibration | 80.64 | 0.950 | 90.18 | 1.050 | 87.86 | 0.976 | 95.08 | 1.019 |

testing | 80.17 | 0.913 | 85.6 | 1.410 | 92.41 | 1.108 | 94.71 | 1.053 | |

Nianyushan | calibration | 75.8 | 0.941 | 83.44 | 1.084 | 84.89 | 0.910 | 85.86 | 1.020 |

testing | 82.38 | 0.803 | 85.39 | 1.329 | 88.30 | 0.939 | 88.39 | 1.077 | |

Gaoguan | calibration | 66.16 | 1.035 | 77.6 | 1.045 | 80.17 | 1.002 | 93.24 | 1.005 |

testing | 76.38 | 0.957 | 77.97 | 0.894 | 80.43 | 0.840 | 89.85 | 0.962 | |

Shimen | calibration | 65.03 | 0.848 | 64.85 | 1.068 | 73.85 | 1.141 | 94.53 | 1.084 |

testing | 72 | 0.772 | 75.72 | 1.281 | 76.90 | 1.089 | 87.99 | 1.055 | |

Tiantang | calibration | 65.47 | 0.985 | 73.06 | 1.049 | 78.08 | 0.960 | 88.66 | 1.131 |

testing | 59.79 | 0.895 | 81.96 | 0.956 | 79.54 | 1.015 | 91.32 | 1.043 | |

Mean | calibration | 69.47 | 0.976 | 78.41 | 1.046 | 82.08 | 1.00 | 92.97 | 1.06 |

testing | 70.16 | 0.879 | 75.86 | 1.155 | 80.62 | 1.04 | 91.52 | 1.07 |

^{2}and WB for the SSA-ANN1 model are 82.08% and 80.62%, and 1.0 and 1.04, during calibration and testing periods, respectively, which are much better than that of the NLPM-ANN model. It means that the reconstructed series obtained by SSA has a strong regularity and is easy to simulate. It also demonstrated that the impact of noise in hydrological time series on model performance is bigger than the seasonal hydrological behavior. Therefore, SSA is an effective way to improve runoff forecasting accuracy. The mean values of R

^{2}for the SSA-ANN2 model are 92.97% and 91.52%, which are much better than those of the SSA-ANN1 model. It is concluded that considering previous runoff as a model input can improve model efficiency greatly.

## 5. Summary and Conclusions

- (1)
- The performance of the ANN model can be improved by data preprocessing techniques. SSA is more effective and it can improve the learning and training ability of the ANN type model significantly. Results also show that the impact of noise in hydrological time series on model performance is bigger than the seasonal hydrological behavior.
- (2)
- Comparing the SSA-ANN1 model with the NLPM-ANN model, the mean values of R
^{2}and WB for the SSA-ANN1 model are 82.08% and 80.62%, and 1.0 and 1.04, during calibration and testing periods, respectively, which are much better than that of the NLPM-ANN model. - (3)
- The SSA-ANN2 model performs best for daily runoff forecasting for all selected watersheds. The effective way for increasing daily runoff forecasting accuracy is to preprocess data series by SSA and select both previous related rainfall and runoff as predictive factors.
- (4)
- There are some limitations in this study. The method to select the contributing components relies on liner correlation analysis, which disregards the existence of nonlinearity in the hydrologic process. The sensitivities and uncertainties of model parameters are not analyzed. All of these will be the focus in our future research.

## Acknowledgment

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Guo, S.; Xiong, L.; Liu, P.; Liu, D.
Daily Runoff Forecasting Model Based on ANN and Data Preprocessing Techniques. *Water* **2015**, *7*, 4144-4160.
https://doi.org/10.3390/w7084144

**AMA Style**

Wang Y, Guo S, Xiong L, Liu P, Liu D.
Daily Runoff Forecasting Model Based on ANN and Data Preprocessing Techniques. *Water*. 2015; 7(8):4144-4160.
https://doi.org/10.3390/w7084144

**Chicago/Turabian Style**

Wang, Yun, Shenglian Guo, Lihua Xiong, Pan Liu, and Dedi Liu.
2015. "Daily Runoff Forecasting Model Based on ANN and Data Preprocessing Techniques" *Water* 7, no. 8: 4144-4160.
https://doi.org/10.3390/w7084144