# Combined Forecasting of Streamflow Based on Cross Entropy

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis of Streamflow Characteristics

^{2}and it is located mainly in the territory of Jilin. According to data obtained at the Wudaogou hydrology station in Huadian City (catchment area of 12,391 km

^{2}), the change in the amplitude of its water level is 7.69 m, the average annual streamflow is 26.4 billion m

^{3}, the average flow is 83.70 m

^{3}/s, the maximum peak discharge is 3010 m

^{3}/s (1975), the minimum flow rate is 0.44 m

^{3}/s (1979), the mean annual sediment volume is 0.48 kg/s, and the annual transportation of sediment is 121 million metric tons.

^{3}in 1995 and 1080 m

^{3}in 2010), the trend and the actual numerical value of the monthly streamflow are relatively strong in each year. Therefore, the selection of reasonably similar years can improve the accuracy of streamflow predictions.

## 3. Data Processing Method

#### 3.1. Data Preprocessing

_{max}is the maximum value in the sample.

#### 3.2. Selecting Similar Years

_{0}and Xm

_{0}:

## 4. Streamflow Forecasting Model Based on CE

#### 4.1. Combined Forecasting Model

_{t}, ${\omega}_{it}$ is the weight of the ith model at time t, and ${\widehat{y}}_{it}$ is the predicted value of the ith model at time t, then the problem of combined forecasting is described as follows:

#### 4.2. The CE Model

_{it}= w

_{0}, set iteration number z = 1;

## 5. Results and Analysis

#### 5.1. Comparison of the Results Obtained with a Single Method

#### 5.2. Comparison with Other Combined Forecasting Models

#### 5.3. Influence of the Historical Data Length on the Prediction Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Zhang, L.X.; Lei, X.Y. Decomposition of time series model in wulasitai River application of annual runoff prediction. J. Water Resour. Water Eng.
**2006**, 17, 22–24. [Google Scholar] - Emili, B.; Alberto, P.; Emilio, S.; Jose, D. Predicting service request in support centers based on nonlinear dynamics ARMA modeling and neural networks. Expert Syst. Appl.
**2008**, 34, 665–672. [Google Scholar] - Li, B.; Yuan, P.; Chang, J. GM (1,1) improved model for predicting annual runoff forecasting. Northeast Water Conserv. Hydroelect.
**2006**, 24, 28–30. [Google Scholar] - Yu, G.R.; Ye, H.; Xie, Z.Q. Projection pursuit auto regression model in predicting runoff of Yangtze River in application. Hohai Univ. J.
**2009**, 37, 263–266. [Google Scholar] - Zhou, H.F.; Li, C. The main stream of the Yellow River Runoff of transient components and frequency analysis and its prediction. Meteor. Sci.
**2003**, 23, 201–207. [Google Scholar] - Qiu, L.; An, K.J.; Wang, W.C. Classification of runoff prediction model based on Markov Bayes. Water Conserv. Sci. Technol. Econ.
**2011**, 17, 1–4. [Google Scholar] - Wang, Q.H.; Qian, X.; Zhang, Y.C. Application of BP neural network in the prediction of runoff and stream reservoir runoff forecast. Environ. Protect. Sci.
**2010**, 23, 19–23. [Google Scholar] - Muhsin, N.; Sinan, U.; Irfan, Y. Side-by-side comparison of horizontal surface flow and free water surface flow constructed wet lands and artificial neural network(ANN)modeling approach. Ecol. Eng.
**2009**, 35, 1255–1263. [Google Scholar] - Yun, W.; Guo, S.L.; Xiong, L.H.; Liu, P.; Liu, D. Daily Runoff Forecasting Model Based on ANN and Data Preprocessing Techniques. Water
**2015**, 7, 4144–4160. [Google Scholar] - Hao, Z.C.; Hao, F.H.; Singh, V.P. A general framework for multivariate multi-index drought prediction based on Multivariate Ensemble Streamflow Prediction (MESP). J. Hydrol.
**2016**, 539, 1–10. [Google Scholar] [CrossRef] - Masselot, P.; Dabo-Niang, S.; Chebana, F.; Ouarda, T.B.M.J. Streamflow forecasting using functional regression. J. Hydrol.
**2016**, 538, 754–766. [Google Scholar] [CrossRef] - Arsenault, R.; Poissant, D.; Brissette, F. Parameter dimensionality reduction of a conceptual model for streamflow prediction in Canadian, snowmelt dominated ungauged basins. Adv. Water Resour.
**2015**, 85, 27–44. [Google Scholar] [CrossRef] - Bates, J.; Granger, C. The combination of forecast. Oper. Res. Quart.
**1969**, 20, 451–468. [Google Scholar] [CrossRef] - Gu, H.Y. Prediction of River Runoff. Master’s Thesis, Northeast Forestry University, Harbin, China, June 2008. [Google Scholar]
- Fan, Y.; Li, Y. Annual runoff combination forecasting method research and application. North Water Conserv. Hydrol. Power
**2006**, 24, 23–27. [Google Scholar] - Yin, J.X.; Jiang, Y.Z.; Lu, F. Based on combination forecasting model of long-term forecasting of reservoir runoff. People’s Yellow River
**2008**, 30, 28–32. [Google Scholar] - Su, X.H. Study on the Short-Term Load Forecasting Based on Artificial Neural Network. Master’s Thesis, Chongqing University, Chongqing, China, May 2005. [Google Scholar]
- Singh, V.P. Entropy Theory and Its Applications in Environmental and Water Engineering; John Wiley: New York, NY, USA, 2013; p. 662. [Google Scholar]
- Marini, G.; De Martino, G.; Fontana, N.; Fiorentino, M.; Singh, V.P. Entropy approach for 2D velocity distribution in open-channel flow. J. Hydraul. Res.
**2011**, 49, 784–790. [Google Scholar] [CrossRef] - Fontana, N.; Marini, G.; De Paola, F. Experimental assessment of a 2-D entropy-based model for velocity distribution in open channel flow. Entropy
**2013**, 15, 988–998. [Google Scholar] [CrossRef] [Green Version] - Li, R.; Liu, H.L.; Lu, Y.; Han, B. A combination method for distribution transformer life prediction based on cross entropy theory. Power Syst. Protect. Control
**2014**, 42, 97–101. [Google Scholar] - Chen, N.; Sha, Q.; Tang, Y.; Oi, Y.; Zhu, L. A Combination Method for Wind Power Predication Based on Cross Entropy Theory. Proc. CSEE
**2012**, 32, 29–34. [Google Scholar] - Liu, J.; Nie, C.X.; Xie, Q. Changes of rainfall runoff relationship and the reasons for the status quo analysis. Anhui Agric. Sci.
**2010**, 38, 5170–5172. [Google Scholar] - Liu, H.W. The Feature Selection Algorithm Based on Information Entropy. Master’s Thesis, Jilin University, Jilin, China, June 2010. [Google Scholar]
- Box, G.E.; Jenkins, G.M. Time Series Analysis: Forecasting and Control, revised ed.; Holden Day: San Francisco, CA, USA, 1976; pp. 80–145. [Google Scholar]
- Si, Q. The Gray Prediction Model of Equal Dimension and New Information and the Forecasting Precision—Based on the Analysis and Prediction of the Pension Insurance for Urban Residents of China. Stat. Thinktank
**2008**, 12, 13–19. (In Chinese) [Google Scholar] - Pieter-tjerk, D.B.; Kroese, D.P.; Shie, M.; Rubinstein, R.Y. A Tutorial on the Cross-Entropy Method. Annal. Oper. Res.
**2005**, 134, 19–67. [Google Scholar]

**Figure 2.**Annual streamflow, annual rainfall, and the ratio of the two curves at Wudaogou station during 1965–2010.

**Figure 4.**Forecast annual streamflow curve. (

**a**) Annual forecast results in 2009; (

**b**) Annual forecast results in 2010.

Year | ARIMA | GM | RBF | CE | ||||
---|---|---|---|---|---|---|---|---|

MRPE | RMSE | MRPE | RMSE | MRPE | RMSE | MRPE | RMSE | |

2006 | 10.95% | 6.96% | 10.55% | 4.79% | 10.58% | 3.98% | 10.53% | 3.67% |

2007 | 10.66% | 5.77% | 9.23% | 5.61% | 9.07% | 4.89% | 8.95% | 5.01% |

2008 | 9.71% | 6.01% | 10.12% | 5.85% | 9.25% | 5.28% | 9.77% | 5.32% |

2009 | 9.89% | 5.25% | 10.35% | 6.08% | 7.32% | 5.01% | 7.24% | 4.99% |

2010 | 11.27% | 7.38% | 11.59% | 8.91% | 8.56% | 7.97% | 8.30% | 6.67% |

Average | 10.50% | 6.27% | 10.37% | 6.25% | 8.96% | 5.43% | 8.96% | 5.13% |

Year | EW | RM | CE | |||
---|---|---|---|---|---|---|

MPE | RMSE | MPE | RMSE | MPE | RMSE | |

2006 | 10.76% | 3.75% | 10.53% | 3.71% | 10.53% | 3.67% |

2007 | 9.23% | 5.56% | 9.14% | 5.34% | 8.95% | 5.01% |

2008 | 9.74% | 5.30% | 9.78% | 5.29% | 9.77% | 5.32% |

2009 | 7.55% | 5.09% | 7.43% | 5.25% | 7.24% | 4.99% |

2010 | 10.07% | 7.24% | 9.52% | 7.08% | 8.30% | 6.67% |

Average | 9.47% | 5.39% | 9.28% | 5.33% | 8.96% | 5.13% |

Case | RBF | RM | CE | |||
---|---|---|---|---|---|---|

MPE | RMSE | MPE | RMSE | MPE | RMSE | |

Case 1 | 9.01% | 5.62% | 9.31% | 5.37% | 8.98% | 5.14% |

Case 2 | 9.37% | 5.71% | 9.42% | 5.43% | 9.11% | 5.25% |

Case 3 | 12.94% | 7.87% | 11.28% | 7.01% | 10.50% | 6.62% |

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**MDPI and ACS Style**

Men, B.; Long, R.; Zhang, J.
Combined Forecasting of Streamflow Based on Cross Entropy. *Entropy* **2016**, *18*, 336.
https://doi.org/10.3390/e18090336

**AMA Style**

Men B, Long R, Zhang J.
Combined Forecasting of Streamflow Based on Cross Entropy. *Entropy*. 2016; 18(9):336.
https://doi.org/10.3390/e18090336

**Chicago/Turabian Style**

Men, Baohui, Rishang Long, and Jianhua Zhang.
2016. "Combined Forecasting of Streamflow Based on Cross Entropy" *Entropy* 18, no. 9: 336.
https://doi.org/10.3390/e18090336