Using a representative station as an example, the daily lake level and river discharge data in a typical normal year (2007) are first selected. Statistical characteristics of the input and the output variables are shown in Table 3. After the normalization procedure, the input and output data are randomly divided into a training group (70% data) and a testing group (30% data). The training data are used to establish the regression relationship and calibrate the critical parameters (Table 4). The site-specific relationship models are successfully developed at all stations, and the performance is shown in Table 5. After that, the testing data are applied to quantitatively validate the model performance. The scatter plots of observed and predicted lake levels, for example, at 3 representative stations are shown in
Figure 7 together with the coefficients of determination. In general, they all show good agreement of the model results with the measured values: at Lujiao in the eastern region,
R2 = 0.9930; at Yangliutan in the southern region,
R2 = 0.9963; and at Xiaohezui in the western,
R2 = 0.9933. The relationship models are also successfully developed at all stations for the typical dry year (2006), with all mean squared errors (MSEs) less than 0.01 and
R2 larger than 0.99. The merit of the SVR-based relationship models is that they may forecast the response of lake levels at different stations accurately and quickly, thus providing an efficient predictor for optimization modeling.
Table 3Statistics of the input (m3/s) and output (m) variables in the relationship model
Input index
. | TGR (x1)
. | Qingjiang (x2)
. | Xiangjiang (x3)
. | Zishui (x4)
. | Yuanjiang (x5)
. | Lishui (x6)
. |
---|
Mean value | 15,889 | 392 | 2,025 | 797 | 2,012 | 434 |
Minimum | 3,210 | 77 | 287 | 160 | 333 | 7 |
Maximum | 61,000 | 889 | 8,420 | 8,130 | 17,600 | 3,820 |
Standard deviation | 13,380 | 239 | 1,644 | 801 | 2,102 | 591 |
Output index
. | Lujiao
. | Yingtian
. | Yangliutan
. | Nanzui
. | Xiaohezui
. | –
. |
---|
Mean value | 28.24 | 28.48 | 28.57 | 30.96 | 31.24 | – |
Minimum | 26.04 | 26.06 | 26.32 | 28.74 | 29.67 | – |
Maximum | 31.97 | 32.15 | 32.16 | 34.85 | 34.57 | – |
Standard deviation | 1.58 | 0.085 | 1.56 | 0.083 | 1.13 | – |
Input index
. | TGR (x1)
. | Qingjiang (x2)
. | Xiangjiang (x3)
. | Zishui (x4)
. | Yuanjiang (x5)
. | Lishui (x6)
. |
---|
Mean value | 15,889 | 392 | 2,025 | 797 | 2,012 | 434 |
Minimum | 3,210 | 77 | 287 | 160 | 333 | 7 |
Maximum | 61,000 | 889 | 8,420 | 8,130 | 17,600 | 3,820 |
Standard deviation | 13,380 | 239 | 1,644 | 801 | 2,102 | 591 |
Output index
. | Lujiao
. | Yingtian
. | Yangliutan
. | Nanzui
. | Xiaohezui
. | –
. |
---|
Mean value | 28.24 | 28.48 | 28.57 | 30.96 | 31.24 | – |
Minimum | 26.04 | 26.06 | 26.32 | 28.74 | 29.67 | – |
Maximum | 31.97 | 32.15 | 32.16 | 34.85 | 34.57 | – |
Standard deviation | 1.58 | 0.085 | 1.56 | 0.083 | 1.13 | – |
Table 4Critical parameters for the relationship model
Parameter
. | Lujiao
. | Yingtian
. | Yangliutan
. | Nanzui
. | Xiaohezui
. |
---|
Error threshold (ε) | 3.82 × 10−4 | 5.63 × 10−7 | 1.90 × 10−4 | 3.17 × 10−6 | 5.65 × 10−2 |
Regularization parameter (C) | 13.4114 | 11.1012 | 5.2005 | 36.8999 | 12,968.4523 |
Kernel parameter (γ) | 0.17279 | 0.21613 | 0.26582 | 0.12718 | 0.058089 |
Parameter
. | Lujiao
. | Yingtian
. | Yangliutan
. | Nanzui
. | Xiaohezui
. |
---|
Error threshold (ε) | 3.82 × 10−4 | 5.63 × 10−7 | 1.90 × 10−4 | 3.17 × 10−6 | 5.65 × 10−2 |
Regularization parameter (C) | 13.4114 | 11.1012 | 5.2005 | 36.8999 | 12,968.4523 |
Kernel parameter (γ) | 0.17279 | 0.21613 | 0.26582 | 0.12718 | 0.058089 |
Table 5Performance of the training and testing for the relationship model
Performance (training)
. | Lujiao
. | Yingtian
. | Yangliutan
. | Nanzui
. | Xiaohezui
. |
---|
Coefficient of determination | 0.998847 | 0.998955 | 0.998590 | 0.999900 | 0.998356 |
Correlation coefficient | 0.999441 | 0.999493 | 0.999315 | 0.999951 | 0.999178 |
Root-mean-square error (m) | 0.053918 | 0.051663 | 0.057776 | 0.015482 | 0.045924 |
Mean absolute error (m) | 0.014989 | 0.013051 | 0.017115 | 0.004078 | 0.0424472 |
Performance (testing)
. | Lujiao
. | Yingtian
. | Yangliutan
. | Nanzui
. | Xiaohezui
. |
---|
Coefficient of determination | 0.993006 | 0.996524 | 0.996286 | 0.997902 | 0.993348 |
Correlation coefficient | 0.996617 | 0.998356 | 0.998177 | 0.998988 | 0.997165 |
Root-mean-square error (m) | 0.124307 | 0.091084 | 0.09300 | 0.069147 | 0.084161 |
Mean absolute error (m) | 0.076960 | 0.059425 | 0.057600 | 0.042857 | 0.063450 |
Performance (training)
. | Lujiao
. | Yingtian
. | Yangliutan
. | Nanzui
. | Xiaohezui
. |
---|
Coefficient of determination | 0.998847 | 0.998955 | 0.998590 | 0.999900 | 0.998356 |
Correlation coefficient | 0.999441 | 0.999493 | 0.999315 | 0.999951 | 0.999178 |
Root-mean-square error (m) | 0.053918 | 0.051663 | 0.057776 | 0.015482 | 0.045924 |
Mean absolute error (m) | 0.014989 | 0.013051 | 0.017115 | 0.004078 | 0.0424472 |
Performance (testing)
. | Lujiao
. | Yingtian
. | Yangliutan
. | Nanzui
. | Xiaohezui
. |
---|
Coefficient of determination | 0.993006 | 0.996524 | 0.996286 | 0.997902 | 0.993348 |
Correlation coefficient | 0.996617 | 0.998356 | 0.998177 | 0.998988 | 0.997165 |
Root-mean-square error (m) | 0.124307 | 0.091084 | 0.09300 | 0.069147 | 0.084161 |
Mean absolute error (m) | 0.076960 | 0.059425 | 0.057600 | 0.042857 | 0.063450 |
Figure 7
Scatter plots of observed and predicted lake levels at 3 representative stations: Lujiao (the eastern), Yangliutan (the southern) and Xiaohezui (the western).
Figure 7
Scatter plots of observed and predicted lake levels at 3 representative stations: Lujiao (the eastern), Yangliutan (the southern) and Xiaohezui (the western).