# An Hourly Streamflow Forecasting Model Coupled with an Enforced Learning Strategy

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed Methodology

#### 2.1. Neural Networks

#### 2.1.1. Back Propagation Neural Network

#### 2.1.2. Radial Basis Function Neural Network

#### 2.1.3. Support Vector Machine

**w**and $b$ are weights and bias, respectively, and can be estimated by minimizing the following structural risk function:

#### 2.1.4. Self-Organizing Map

#### 2.2. Enforced Learning Strategy

## 3. Application and Result Discussion

#### 3.1. The Study Area and Data

^{2}, ranks 4th in Taiwan. The length of the main river is 119 km, and the average slope is 1/92. In this basin, floods caused by heavy rainfalls are quite common. The metropolis of Taichung, which is a major city with a population of about three million in central Taiwan, is located downstream in the study area. Therefore, an accurate, efficient and robust flow forecasting model is needed for the study area.

#### 3.2. Input Design and Parameter Setting of NN-Base Models

Lag Length $j$ | Pearson Product-Moment Correlation Coefficient $\text{\rho}$ | |
---|---|---|

Between ${Q}_{t+1}$ and ${R}_{t-j}$ | Between ${Q}_{t+1}$ and ${Q}_{t-j}$ | |

0 | 0.48 | 0.95 |

1 | 0.55 | 0.88 |

2 | 0.58 | 0.80 |

3 | 0.56 | 0.72 |

4 | 0.52 | 0.65 |

5 | 0.46 | 0.58 |

#### 3.3. Cross Validation and Performance Measures

_{p}) is used. For a single typhoon event, the ET

_{p}is written as:

_{p}(MET

_{p}) is calculated. Moreover, the percentage error of peak flow (EQ

_{p}) is used. For a single typhoon event, the EQ

_{p}is written as:

_{p}(MEQ

_{p}) is calculated.

_{p}, and EQ

_{p}are used to measure the forecasting error related to the peak values. Moreover, since the cross-validation test is used in this paper, the mean values of these four criteria (i.e., MRRMSE, MCE, MET

_{p}, and MEQ

_{p}) are further used to compare the forecasting performance of different NNs. Hence, based on the use of these criteria, a just conclusion is expected to be reached.

#### 3.4. Performance Comparisons among Four NN-Based Models

_{p}

_{,}and MEQ

_{p}values increase with increasing forecast lead time, and the MCE values decrease with increasing forecast lead time. It is observed that the SOLO and the SVM models yield lower MRRMSE values and higher CE values than the BPN and the RBFN models for 1- to 3-h ahead forecasting. The results indicate the SOLO and the SVM models perform better than the BPN and the RBFN models. As to the comparison between the SOLO and the SVM models, the SOLO model performs better than the SVM model for 1-h ahead forecasting. For 2-h ahead forecasting, these two models perform equally well, and for 3-h ahead forecasting the SVM model performs better than the SOLO model. It may be speculated that for 1-h ahead forecasting the relation between rainfall and flow is slightly nonlinear, and hence the piecewise linear model (i.e., SOLO) quickly captures the relationship hidden in the training data and yields better forecasts as compared to the nonlinear model (i.e., SVM). For 3-h ahead forecasting the relation between rainfall and flow is very complicated and highly nonlinear, and therefore the SVM model performs better than the SOLO model. As to the MET

_{p}and MEQ

_{p}values, the SOLO model yields the lowest MET

_{p}, while the SVM gives the lowest MEQ

_{p}. Overall, it is concluded that the forecasts resulting from the SOLO and the SVM models are more accurate than those from the other two models in this study. Among these four models, the forecasting performance of the RBFN model is the worst. For 3-h ahead forecasting, the CE value from the RBFN model is even negative. That indicates the observed mean is a better forecast than the output of the RBFN model. Maybe the random selection procedure used herein cannot effectively obtain the best center vectors of RBFN hidden neurons.

Model | MRRMSE (%) | MCE | MET_{p} (h) | MEQ_{P} (%) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1-h ahead | 2-h ahead | 3-h ahead | 1-h ahead | 2-h ahead | 3-h ahead | 1-h ahead | 2-h ahead | 3-h ahead | 1-h ahead | 2-h ahead | 3-h ahead | |

BPN | 26.5 | 32.2 | 44.5 | 0.83 | 0.57 | 0.15 | 2.1 | 2.8 | 3.9 | 10.6 | 18.2 | 28.1 |

RBFN | 15.4 | 32.0 | 42.2 | 0.87 | 0.48 | −0.11 | 2.2 | 2.8 | 4.5 | 9.9 | 24.6 | 37.7 |

SOLO | 9.1 | 19.0 | 28.8 | 0.94 | 0.72 | 0.34 | 0.7 | 1.3 | 2.2 | 8.8 | 16.5 | 23.1 |

SVM | 12.1 | 18.8 | 27.5 | 0.90 | 0.72 | 0.47 | 2.1 | 2.8 | 2.7 | 7.2 | 12.9 | 18.6 |

_{p}is the mean error of time to peak flow; MEQ

_{p}is the mean error of peak flow.

Lead Time (h) | CV (%) | |||
---|---|---|---|---|

BPN | RBFN | SOLO | SVM | |

1 | 0.1 | 2.4 | 0 | 0 |

2 | 0.3 | 23.1 | 0 | 0 |

3 | 16.1 | −151.7 | 0 | 0 |

#### 3.5. Effects of the Enforced Learning Strategy on NN-Based Models

#### 3.5.1. Comparison of the SOLO Models with and without the Enforced Learning Strategy

_{p}values from the enforced SOLO model are lower than those from the SOLO model. As to MET

_{p}, the performance of the SOLO and the enforced SOLO models are the same. That is, the enforced SOLO model also provides more accurate forecasts for the peak flow.

_{P}values of the SOLO model are 12%, 27% and 55% for 1- to 3-h ahead forecasting. By using the enforced SOLO model, these corresponding EQ

_{P}values are 6%, 11% and 37%. A significant improvement in reducing the error of peak flow forecasting is clearly observed. Hence, according to the comparison results above, it is clearly concluded that the improved streamflow forecasts are indeed obtained by the enforced SOLO model (i.e., the SOLO model with the enforced learning strategy).

**Figure 5.**Performance comparison of the SOLO and the enforced SOLO models: (

**a**) MRRMSE; (

**b**) MCE; (

**c**) MET

_{p}and (

**d**) MEQ

_{p}.

#### 3.5.2. Comparison of the SVM Models with and without the Enforced Learning Strategy

_{p}and MEQ

_{p}values from the enforced SVM model are lower than those from the SVM model. It is concluded that the enforced SVM model indeed improves the forecasts of overall flows as well as the peaks, and the enforced learning strategy successfully improves the forecasting performance of the SVM model.

**Figure 6.**Observed flows versus corresponding forecasts resulting from the SOLO and the enforced SOLO models: (

**a**) 1-h ahead; (

**b**) 2-h ahead and (

**c**) 3-h ahead.

_{P}values of the SVM model are 30%, 49% and 58% for 1- to 3-h ahead forecasting. Those are reduced to 6%, 16% and 42% by means of the enforced SVM. Again, a significant improvement in reducing the error of peak flow forecasting is clearly observed. Hence, based on the results above, it is concluded that the improved forecasts are indeed obtained by using the enforced SVM model (i.e., the SVM model with the enforced learning strategy).

**Figure 7.**Performance comparison of the SVM and the enforced SVM models: (

**a**) MRRMSE; (

**b**) MCE; (

**c**) MET

_{p}; and (

**d**) MEQ

_{P}.

**Figure 8.**Observed flows versus corresponding forecasts resulting from the SVM and the enforced SVM models: (

**a**) 1-h ahead; (

**b**) 2-h ahead and (

**c**) 3-h ahead.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wu, M.-C.; Lin, G.-F. An Hourly Streamflow Forecasting Model Coupled with an Enforced Learning Strategy. *Water* **2015**, *7*, 5876-5895.
https://doi.org/10.3390/w7115876

**AMA Style**

Wu M-C, Lin G-F. An Hourly Streamflow Forecasting Model Coupled with an Enforced Learning Strategy. *Water*. 2015; 7(11):5876-5895.
https://doi.org/10.3390/w7115876

**Chicago/Turabian Style**

Wu, Ming-Chang, and Gwo-Fong Lin. 2015. "An Hourly Streamflow Forecasting Model Coupled with an Enforced Learning Strategy" *Water* 7, no. 11: 5876-5895.
https://doi.org/10.3390/w7115876