# An EMD-Based Chaotic Least Squares Support Vector Machine Hybrid Model for Annual Runoff Forecasting

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Study Area and Data

^{2}with a river length of 716 km, as seen in Figure 1. With a semi-arid climate, the Fenhe River basin has an average annual precipitation ranging from 300 to 750 mm and an annual mean temperature varying from 9 to 12 °C. The upper reaches of the Fenhe River are in the area above Lancun station. Annual runoff data from four hydrologic stations in the upper reaches, i.e., Shangjingyou, Fenhe reservoir, Zhaishang, and Lancun, were analyzed. Shangjingyou station lies in Lan River, which belongs to one of the tributaries of Fenhe River. The Zhaishang, Fenhe reservoir, and Lancun stations are in the mainstream of Fenhe River. The locations of the stations are shown in Figure 1. These runoff data of flow records covering 1956 to 2000 were obtained from the government hydrologic database. The first 40 years (1956–1995) of runoff data were applied for calibration purpose and the remaining five years of data were used for validation.

#### 2.2. Methods

#### 2.2.1. Empirical Mode Decomposition

- Step 1)
- Identify all the extreme (maxima and minima) values of a time series $x(t)$.
- Step 2)
- Generate the upper and lower envelope ($u(t)$ and $v(t)$) by applying cubic spline interpolation.
- Step 3)
- Compute a local mean curve $m(t)$ of the two envelopes at the same time using$$m(t)=(u(t)+v(t))/2.$$
- Step 4)
- Calculate the difference,$$h(t)=x(t)-m(t).$$
- Step 5)
- Check the properties of $h(t)$; it will be considered a valid IMF if it satisfies these two conditions:
- The number of extreme and zero crossings must be equal or differ at most by one.
- At any point, the local mean value of the envelope defined by the local extremes must be zero.

- Step 6)
- When $h(t)$ is not qualified as an IMF, repeat Steps 1) to 5) by sifting the residual series. The sifting process stops when the residual (i.e., the trend term; $r(t)$) satisfies the predetermined criteria.

#### 2.2.2. Phase Space Reconstruction and Chaotic Characteristics Identification

#### 2.2.3. Chaotic Least Squares Support Vector Machine Model

## 3. Model Results and Discussion

#### 3.1. Stationarity Test of the Original Runoff Series

#### 3.2. EMD of Runoff Series

#### 3.3. Determination of Delay Time τ and Embedding Dimension m

#### 3.4. Identification of Chaotic Characteristics

_{1}of IMFs for Shangjingyou station is greater than zero, except for IMF5, which indicates that IMFs have chaotic characteristics. The average period of the IMF5 for Shangjingyou station was 23 years, but the lengths of IMF5 for Shangjingyou station are only 45 years. Therefore, the Lyapunov exponent method would be unable to analyze the chaotic characteristics of the IMF5. In view of this, the Power Spectrum method is used to analyze the chaotic characteristics of the IMF5 for Shangjingyou station. Figure 4 shows that the IMF5 for Shangjingyou station has a single peak, which indicates that the IMF5 time series does not have chaotic characteristics.

#### 3.5. Hybrid Model

#### 3.6. Comparative Analysis

_{1}of the IMFs for Fenhe reservoir, Zhaishang, and Lancun stations are more than 0, except for the IMF5 for Fenhe reservoir station and the IMF4 for Zhaishang and Lancun stations, which indicates that the IMFs have chaotic characteristics. The average period of the IMF5 for Fenhe reservoir station and the IMF4 for Zhaishang and Lancun stations is 23 years. Therefore, the Power Spectrum method is used to analyze the chaotic characteristics of the IMF5 for Fenhe reservoir station, and the IMF4 for Zhaishang and Lancun stations. Figure 9, Figure 10 and Figure 11 show that the IMF5 for Fenhe reservoir stations and the IMF4 for Zhaishang and Lancun stations each have single peaks, which indicates that the three time series do not have chaotic characteristics.

#### 3.7. Evaluation of Model Performance

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Delay time τ of several intrinsic mode functions (IMFs) for Shangjingyou station ((

**a**) IMF1; (

**b**) IMF2; (

**c**) IMF3; (

**d**) IMF4; and (

**e**) IMF5).

**Figure 5.**Prediction results of annual runoff for Shangjingyou station. CLSSVM: chaotic least squares support vector machine; EMD-CLSSVM: EMD-based chaotic LSSVM.

**Table 1.**Lagrange Multiplier (LM) statistic, delay time τ, embedding dimension m, average period p, and Lyapunov exponent λ

_{1}of intrinsic mode functions (IMFs) for Shangjingyou station.

IMFs | LM Statistic | τ | m | Average Period p | Lyapunov Exponent λ_{1} |
---|---|---|---|---|---|

IMF1 | 0.114 | 1 | 2 | 3 | 0.113 |

IMF2 | 0.148 | 1 | 5 | 4 | 0.160 |

IMF3 | 0.063 | 2 | 4 | 4 | 0.101 |

IMF4 | 0.075 | 4 | 6 | 8 | 0.040 |

IMF5 | 0.571 | 1 | 6 | 23 | — |

**Table 2.**Least squares support vector machine (LSSVM) model parameters of IMFs for Shangjingyou station. C: penalty factor; σ: parameter of kernel function.

IMFs | IMF1 | IMF2 | IMF3 | IMF4 |
---|---|---|---|---|

C | 625.2 | 316.0 | 68.1 | 18.1 |

σ | 0.1 | 49.2 | 7.2 | 8.3 |

**Table 3.**LM statistic, delay time τ, embedding dimension m, average period p, and Lyapunov exponent λ

_{1}of IMFs for the three stations.

Stations | IMFs | LM Statistic | τ | m | Average Period p | Lyapunov Exponent λ_{1} |
---|---|---|---|---|---|---|

Fenhe reservoir | IMF1 | 0.180 | 1 | 7 | 7 | 0.091 |

IMF2 | 0.087 | 1 | 7 | 7 | 0.042 | |

IMF3 | 0.105 | 3 | 6 | 6 | 0.023 | |

IMF4 | 0.112 | 3 | 5 | 5 | 0.168 | |

IMF5 | 0.363 | 2 | 4 | 23 | — | |

Zhaishang | IMF1 | 0.248 | 1 | 7 | 7 | 0.01 |

IMF2 | 0.119 | 2 | 7 | 7 | 0.169 | |

IMF3 | 0.097 | 2 | 3 | 3 | 0.169 | |

IMF4 | 0.335 | 1 | 4 | 23 | — | |

Lancun | IMF1 | 0.129 | 1 | 7 | 7 | 0.163 |

IMF2 | 0.129 | 2 | 7 | 7 | 0.017 | |

IMF3 | 0.082 | 2 | 4 | 4 | 0.140 | |

IMF4 | 0.329 | 2 | 4 | 23 | — |

**Table 4.**Performance indicators of different models for four hydrologic stations during training and testing periods.

Station | Hybrid Model | Training | Testing | ||||||
---|---|---|---|---|---|---|---|---|---|

QR | RMSE (10^{6} m^{3}) | MARE | MAE (10^{6} m^{3}) | QR | RMSE (10^{6} m^{3}) | MARE | MAE (10^{6} m^{3}) | ||

Shangjingyou | CLSSVM | 61% | 10.19 | 0.21 | 8.12 | 100% | 2.49 | 0.07 | 2.03 |

EMD-CLSSVM | 97% | 3.78 | 0.07 | 2.93 | 100% | 1.52 | 0.05 | 1.51 | |

Fenhe reservoir | CLSSVM | 65% | 120.33 | 0.27 | 70.83 | 100% | 11.03 | 0.05 | 9.89 |

EMD-CLSSVM | 97% | 75.17 | 0.21 | 53.43 | 100% | 9.93 | 0.03 | 7.95 | |

Zhaishang | CLSSVM | 68% | 114.91 | 0.32 | 90.91 | 40% | 77.05 | 0.26 | 67.64 |

EMD-CLSSVM | 97% | 40.19 | 0.09 | 30.33 | 100% | 17.71 | 0.07 | 15.72 | |

Lancun | CLSSVM | 58% | 106.63 | 0.30 | 65.58 | 40% | 85.84 | 0.30 | 74.17 |

EMD-CLSSVM | 97% | 24.18 | 0.07 | 19.11 | 100% | 12.16 | 0.05 | 10.18 |

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**MDPI and ACS Style**

Zhao, X.; Chen, X.; Xu, Y.; Xi, D.; Zhang, Y.; Zheng, X.
An EMD-Based Chaotic Least Squares Support Vector Machine Hybrid Model for Annual Runoff Forecasting. *Water* **2017**, *9*, 153.
https://doi.org/10.3390/w9030153

**AMA Style**

Zhao X, Chen X, Xu Y, Xi D, Zhang Y, Zheng X.
An EMD-Based Chaotic Least Squares Support Vector Machine Hybrid Model for Annual Runoff Forecasting. *Water*. 2017; 9(3):153.
https://doi.org/10.3390/w9030153

**Chicago/Turabian Style**

Zhao, Xuehua, Xu Chen, Yongxin Xu, Dongjie Xi, Yongbo Zhang, and Xiuqing Zheng.
2017. "An EMD-Based Chaotic Least Squares Support Vector Machine Hybrid Model for Annual Runoff Forecasting" *Water* 9, no. 3: 153.
https://doi.org/10.3390/w9030153