River Stage Modeling by Combining Maximal Overlap Discrete Wavelet Transform, Support Vector Machines and Genetic Algorithm
Abstract
:1. Introduction
2. Materials and Methods
2.1. Data Used
2.2. Discrete Wavelet Transform (DWT)
2.3. Maximal Overlap Discrete Wavelet Transform (MODWT)
2.4. Multilayer Perceptron (MLP)
2.5. Support Vector Machine (SVM)
2.6. Genetic Algotrithm (GA)
- Step 1. Generate an initial random population .
- Step 2. Compute the fitness of each chromosome in the population, and assign probability typically proportional to the fitness.
- Step 3. Reproduce new population using selection, crossover and mutation operators.
- Step 4. Repeat from step 2 to step 3 until stop conditions are met. The algorithm yields as the optimum.
2.7. River Stage Modeling Using DWT and MODWT
- Step 1. Decompose original input signals into sub-signals (detail and approximation components) utilizing DWT and MODWT.
- Step 2. Select effective inputs among the sub-signals.
- Step 3. Train and test single models, MLP and SVM, utilizing the effective inputs.
2.8. Model Efficiency Evaluation
- -
- Dimensionless indices: coefficient of efficiency (CE), index of agreement (d) and coefficient of determination (r2)
- -
- Residual error-based indices: root-mean-square error (RMSE), mean absolute error (MAE), mean squared relative error (MSRE) and mean higher order error (MS4E)
2.9. Model Effectiveness Evaluation
3. Results and Discussion
3.1. Model Development
3.2. Model Performance Assessment
3.3. Graphical Comparison
4. Conclusions
- (1)
- For the overall stage, the MODWT–SVM models achieve better efficiency and effectiveness based on the statistical indices, and are more accurate than the single models based on the graphical comparison. For the low, medium and high stages, the MODWT–SVM models perform better than the single models, in terms of efficiency and effectiveness. These results indicate that the conjunction of MODWT, SVM and GA can improve the performance of SVM models and outperform single models in daily river stage modeling.
- (2)
- For the overall stage, the MODWT–SVM models achieve better efficiency and effectiveness based on the statistical indices, and are more accurate than the MODWT–MLP and DWT-based models based on the graphical comparison. For the low and high stages, the MODWT–SVM models performed better than the MODWT–MLP and DWT-based models, in terms of efficiency and effectiveness. For the medium stage, the DWT–MLP models outperform the MODWT–SVM models, in terms of the statistical indices. These results demonstrate that the MODWT–based models using the SVM model can improve model performance and accuracy better than those using the MLP model in daily river stage modeling. Also, hybrid models coupling MODWT, SVM and GA can enhance model performance and accuracy in daily river stage modeling as compared with those combined with DWT.
- (3)
- The MODWT–SVM2–c12 model achieves the best efficiency for the overall, low and high stages, based on the efficiency indices; the MODWT–SVM1–c12 model for the low stage; the DWT–MLP1–d18 model for the medium stage; and the MODWT–SVM1–s18 model for the high stage. Also, the MODWT–SVM1–c12 model achieves the best effectiveness for the overall and low stages; the MODWT–SVM2–c12 model for the overall, low, medium and high stages; and the MODWT–SVM1–s18 model for the high stage. These results indicate that the performance of the MODWT–SVM models is dependent on input combination and mother wavelets. Furthermore, the MODWT–SVM model using the c12 mother wavelet can improve model efficiency and effectiveness in daily river stage modeling. Therefore, the results obtained from this study demonstrate that the conjunction of MODWT, SVM and GA can be an efficient and effective method for modeling daily river stages.
Author Contributions
Conflicts of Interest
Appendix A. Model Efficiency Indices
- Coefficient of efficiency (CE): ,
- Index of agreement (d): ,
- Coefficient of determination (r2): ,
- Root-mean-square error (RMSE): ,
- Mean absolute error (MAE): ,
- Mean squared relative error (MSRE): ,
- Mean higher order error (MS4E): ,
Appendix B. Model Effectiveness Indices
- Average absolute relative error (AARE): ,
- Threshold statistics (TS): ,
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Input Sets | Input Variables | Output Variables |
---|---|---|
Set 1 | SSiC(t-6), SSiC(t-5), SSiC(t-4), SSiC(t-3), SSiC(t-2), SSiC(t-1), SSoC(t-1), SSoC(t) | SSiC(t) |
Set 2 | SSiC(t-6), SSiC(t-5), SSiC(t-4), SSiC(t-3), SSiC(t-2), SSiC(t-1), SSoC(t-1), SSoC(t), SBH(t-1), SBH(t) | SSiC(t) |
Set 3 | SSiC(t-6), SSiC(t-5), SSiC(t-4), SSiC(t-3), SSiC(t-2), SSiC(t-1), SSoC(t-1), SSoC(t), SBH(t-1), SBH(t), SHG(t-1), SHG(t) | SSiC(t) |
Models | CE | d | r2 | RMSE (m) | MAE (m) | MSRE (10−5) | MS4E (10−6 m4) | AARE (%) | TS0.01 (%) | TS0.02 (%) | TS0.05 (%) | TS0.10 (%) | TS0.50 (%) | TS1.00 (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
MLP3 | 0.961 | 0.990 | 0.961 | 0.0419 | 0.0243 | 492.090 | 99.187 | 0.025 | 35.7 | 62.8 | 88.3 | 96.2 | 99.9 | 100.0 |
SVM2 | 0.969 | 0.992 | 0.969 | 0.0374 | 0.0196 | 357.683 | 98.212 | 0.026 | 35.0 | 61.7 | 87.5 | 95.8 | 99.9 | 100.0 |
DWT-MLP1-d18 | 0.996 | 0.999 | 0.996 | 0.0129 | 0.0089 | 60.130 | 0.482 | 0.009 | 68.0 | 92.1 | 99.4 | 99.8 | 100.0 | 100.0 |
DWT-MLP1-s6 | 0.996 | 0.999 | 0.996 | 0.0130 | 0.0089 | 54.750 | 0.765 | 0.009 | 68.4 | 91.7 | 99.4 | 99.9 | 100.0 | 100.0 |
DWT-MLP1-s18 | 0.995 | 0.999 | 0.995 | 0.0153 | 0.0097 | 91.610 | 1.208 | 0.010 | 68.3 | 88.3 | 99.1 | 99.8 | 100.0 | 100.0 |
DWT-SVM3-c12 | 0.990 | 0.997 | 0.991 | 0.0211 | 0.0117 | 125.750 | 9.914 | 0.012 | 63.2 | 85.3 | 96.9 | 99.5 | 100.0 | 100.0 |
DWT-SVM1-s18 | 0.989 | 0.997 | 0.990 | 0.0218 | 0.0117 | 119.140 | 9.382 | 0.012 | 64.2 | 84.0 | 97.2 | 99.3 | 100.0 | 100.0 |
DWT-SVM2-s18 | 0.989 | 0.997 | 0.990 | 0.0221 | 0.0114 | 124.370 | 11.197 | 0.012 | 65.4 | 84.5 | 97.3 | 99.3 | 100.0 | 100.0 |
MODWT-MLP3-s6 | 0.993 | 0.998 | 0.993 | 0.0177 | 0.0119 | 91.980 | 3.250 | 0.012 | 53.8 | 85.5 | 98.7 | 99.7 | 100.0 | 100.0 |
MODWT-MLP2-d6 | 0.993 | 0.998 | 0.993 | 0.0178 | 0.0114 | 94.810 | 2.051 | 0.012 | 60.4 | 84.7 | 98.0 | 99.6 | 100.0 | 100.0 |
MODWT-MLP3-c6 | 0.993 | 0.998 | 0.993 | 0.0178 | 0.0109 | 93.410 | 2.035 | 0.011 | 64.6 | 85.6 | 97.8 | 99.7 | 100.0 | 100.0 |
MODWT-SVM2-c12 | 0.997 | 0.999 | 0.997 | 0.0113 | 0.0049 | 29.540 | 1.430 | 0.005 | 90.1 | 97.6 | 99.3 | 99.5 | 100.0 | 100.0 |
MODWT-SVM1-c12 | 0.997 | 0.999 | 0.997 | 0.0118 | 0.0048 | 31.020 | 1.879 | 0.005 | 91.6 | 97.2 | 99.1 | 99.7 | 100.0 | 100.0 |
MODWT-SVM1-s18 | 0.997 | 0.999 | 0.997 | 0.0115 | 0.0068 | 32.530 | 0.657 | 0.007 | 78.7 | 94.3 | 99.5 | 99.8 | 100.0 | 100.0 |
Models | CE | d | r2 | RMSE (m) | MAE (m) | MSRE (10−5) | MS4E (10−6 m4) | AARE (%) | TS0.01 (%) | TS0.02 (%) | TS0.05 (%) | TS0.10 (%) | TS0.50 (%) | TS1.00 (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
MLP3 | 0.797 | 0.943 | 0.808 | 0.0313 | 0.0195 | 0.0103 | 10.772 | 0.0200 | 43.1 | 68.6 | 91.3 | 98.0 | 100.0 | 100.0 |
SVM2 | 0.770 | 0.936 | 0.785 | 0.0332 | 0.0209 | 0.0116 | 13.287 | 0.0214 | 41.8 | 66.8 | 89.9 | 97.3 | 100.0 | 100.0 |
DWT-MLP1-d18 | 0.970 | 0.992 | 0.972 | 0.0120 | 0.0087 | 0.0015 | 0.133 | 0.0090 | 68.3 | 92.0 | 99.7 | 100.0 | 100.0 | 100.0 |
DWT-MLP1-s6 | 0.973 | 0.993 | 0.976 | 0.0114 | 0.0088 | 0.0014 | 0.086 | 0.0090 | 67.2 | 92.5 | 99.8 | 100.0 | 100.0 | 100.0 |
DWT-MLP1-s18 | 0.954 | 0.988 | 0.954 | 0.0148 | 0.0097 | 0.0023 | 0.625 | 0.0100 | 67.2 | 87.7 | 99.1 | 99.8 | 100.0 | 100.0 |
DWT-SVM3-c12 | 0.952 | 0.987 | 0.952 | 0.0152 | 0.0094 | 0.0024 | 0.930 | 0.0096 | 69.4 | 88.6 | 98.3 | 99.8 | 100.0 | 100.0 |
DWT-SVM1-s18 | 0.958 | 0.989 | 0.958 | 0.0143 | 0.0085 | 0.0021 | 0.768 | 0.0088 | 73.6 | 89.7 | 98.3 | 99.8 | 100.0 | 100.0 |
DWT-SVM2-s18 | 0.954 | 0.988 | 0.954 | 0.0149 | 0.0085 | 0.0023 | 1.665 | 0.0087 | 74.7 | 89.9 | 98.6 | 99.8 | 100.0 | 100.0 |
MODWT-MLP3-s6 | 0.958 | 0.989 | 0.965 | 0.0142 | 0.0109 | 0.0021 | 0.259 | 0.0112 | 53.4 | 88.0 | 99.4 | 100.0 | 100.0 | 100.0 |
MODWT-MLP2-d6 | 0.959 | 0.989 | 0.960 | 0.0141 | 0.0102 | 0.0021 | 0.271 | 0.0104 | 61.3 | 87.7 | 99.2 | 100.0 | 100.0 | 100.0 |
MODWT-MLP3-c6 | 0.962 | 0.990 | 0.963 | 0.0135 | 0.0089 | 0.0019 | 0.399 | 0.0091 | 69.7 | 90.0 | 99.2 | 99.8 | 100.0 | 100.0 |
MODWT-SVM2-c12 | 0.992 | 0.998 | 0.992 | 0.0063 | 0.0033 | 0.0004 | 0.174 | 0.0034 | 95.8 | 99.4 | 99.7 | 99.8 | 100.0 | 100.0 |
MODWT-SVM1-c12 | 0.992 | 0.998 | 0.992 | 0.0064 | 0.0030 | 0.0004 | 0.164 | 0.0031 | 97.5 | 99.5 | 99.5 | 100.0 | 100.0 | 100.0 |
MODWT-SVM1-s18 | 0.990 | 0.998 | 0.990 | 0.0069 | 0.0047 | 0.0005 | 0.030 | 0.0048 | 88.5 | 98.8 | 99.8 | 100.0 | 100.0 | 100.0 |
Models | CE | d | r2 | RMSE (m) | MAE (m) | MSRE (10−5) | MS4E (10−6 m4) | AARE (%) | TS0.01 (%) | TS0.02 (%) | TS0.05 (%) | TS0.10 (%) | TS0.50 (%) | TS1.00 (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
MLP3 | 0.603 | 0.911 | 0.723 | 0.0385 | 0.0248 | 0.0155 | 29.965 | 0.0253 | 24.7 | 58.7 | 89.4 | 97.5 | 100.0 | 100.0 |
SVM2 | 0.606 | 0.913 | 0.730 | 0.0383 | 0.0246 | 0.0154 | 27.321 | 0.0252 | 25.8 | 58.3 | 89.8 | 97.2 | 100.0 | 100.0 |
DWT-MLP1-d18 | 0.977 | 0.995 | 0.984 | 0.0093 | 0.0069 | 0.0009 | 0.051 | 0.0070 | 77.4 | 96.8 | 100.0 | 100.0 | 100.0 | 100.0 |
DWT-MLP1-s6 | 0.975 | 0.994 | 0.980 | 0.0096 | 0.0066 | 0.0010 | 0.100 | 0.0067 | 80.2 | 95.8 | 99.3 | 100.0 | 100.0 | 100.0 |
DWT-MLP1-s18 | 0.976 | 0.994 | 0.979 | 0.0095 | 0.0066 | 0.0009 | 0.061 | 0.0068 | 80.9 | 95.4 | 100.0 | 100.0 | 100.0 | 100.0 |
DWT-SVM3-c12 | 0.875 | 0.970 | 0.898 | 0.0216 | 0.0124 | 0.0049 | 5.589 | 0.0127 | 58.7 | 87.6 | 96.1 | 99.3 | 100.0 | 100.0 |
DWT-SVM1-s18 | 0.832 | 0.959 | 0.856 | 0.0251 | 0.0129 | 0.0066 | 15.514 | 0.0131 | 56.9 | 84.8 | 97.5 | 98.9 | 100.0 | 100.0 |
DWT-SVM2-s18 | 0.854 | 0.965 | 0.874 | 0.0234 | 0.0121 | 0.0057 | 11.551 | 0.0124 | 58.7 | 85.5 | 97.5 | 98.9 | 100.0 | 100.0 |
MODWT-MLP3-s6 | 0.936 | 0.984 | 0.949 | 0.0154 | 0.0100 | 0.0025 | 1.056 | 0.0103 | 63.3 | 90.1 | 98.9 | 99.6 | 100.0 | 100.0 |
MODWT-MLP2-d6 | 0.946 | 0.987 | 0.951 | 0.0142 | 0.0085 | 0.0021 | 0.989 | 0.0087 | 72.1 | 91.2 | 98.9 | 99.3 | 100.0 | 100.0 |
MODWT-MLP3-c6 | 0.945 | 0.987 | 0.952 | 0.0143 | 0.0094 | 0.0021 | 0.723 | 0.0096 | 66.1 | 88.7 | 99.3 | 99.6 | 100.0 | 100.0 |
MODWT-SVM2-c12 | 0.952 | 0.988 | 0.955 | 0.0134 | 0.0057 | 0.0019 | 2.688 | 0.0058 | 88.3 | 97.2 | 99.3 | 99.3 | 100.0 | 100.0 |
MODWT-SVM1-c12 | 0.950 | 0.988 | 0.953 | 0.0137 | 0.0058 | 0.0020 | 2.914 | 0.0059 | 88.3 | 96.8 | 99.3 | 99.3 | 100.0 | 100.0 |
MODWT-SVM1-s18 | 0.940 | 0.985 | 0.946 | 0.0149 | 0.0083 | 0.0023 | 2.110 | 0.0084 | 71.7 | 93.6 | 99.3 | 99.3 | 100.0 | 100.0 |
Models | CE | d | r2 | RMSE (m) | MAE (m) | MSRE (10−5) | MS4E (10−6 m4) | AARE (%) | TS0.01 (%) | TS0.02 (%) | TS0.05 (%) | TS0.10 (%) | TS0.50 (%) | TS1.00 (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
MLP3 | 0.921 | 0.979 | 0.921 | 0.0850 | 0.0518 | 0.0741 | 818.660 | 0.0527 | 20.8 | 38.7 | 67.0 | 82.1 | 99.1 | 100.0 |
SVM2 | 0.927 | 0.981 | 0.928 | 0.0814 | 0.0502 | 0.0679 | 670.961 | 0.0511 | 17.9 | 39.6 | 67.0 | 83.0 | 99.1 | 100.0 |
DWT-MLP1-d18 | 0.994 | 0.999 | 0.996 | 0.0229 | 0.0152 | 0.0054 | 3.743 | 0.0155 | 40.6 | 80.2 | 96.2 | 98.1 | 100.0 | 100.0 |
DWT-MLP1-s6 | 0.993 | 0.998 | 0.995 | 0.0245 | 0.0159 | 0.0061 | 6.646 | 0.0162 | 44.3 | 76.4 | 97.2 | 99.1 | 100.0 | 100.0 |
DWT-MLP1-s18 | 0.992 | 0.998 | 0.993 | 0.0266 | 0.0175 | 0.0072 | 7.794 | 0.0177 | 41.5 | 72.6 | 97.2 | 99.1 | 100.0 | 100.0 |
DWT-SVM3-c12 | 0.981 | 0.995 | 0.987 | 0.0410 | 0.0237 | 0.0171 | 75.787 | 0.0241 | 37.7 | 59.4 | 90.6 | 98.1 | 100.0 | 100.0 |
DWT-SVM1-s18 | 0.981 | 0.995 | 0.984 | 0.0412 | 0.0272 | 0.0173 | 45.099 | 0.0277 | 26.4 | 47.2 | 89.6 | 97.2 | 100.0 | 100.0 |
DWT-SVM2-s18 | 0.978 | 0.994 | 0.984 | 0.0443 | 0.0270 | 0.0201 | 67.891 | 0.0275 | 27.4 | 49.1 | 88.7 | 97.2 | 100.0 | 100.0 |
MODWT-MLP3-s6 | 0.987 | 0.997 | 0.987 | 0.0348 | 0.0224 | 0.0124 | 27.195 | 0.0227 | 31.1 | 58.5 | 94.3 | 98.1 | 100.0 | 100.0 |
MODWT-MLP2-d6 | 0.985 | 0.996 | 0.987 | 0.0365 | 0.0263 | 0.0137 | 15.645 | 0.0268 | 23.6 | 49.1 | 87.7 | 98.1 | 100.0 | 100.0 |
MODWT-MLP3-c6 | 0.984 | 0.996 | 0.986 | 0.0378 | 0.0269 | 0.0147 | 15.432 | 0.0274 | 29.2 | 50.9 | 84.9 | 99.1 | 100.0 | 100.0 |
MODWT-SVM2-c12 | 0.994 | 0.999 | 0.995 | 0.0230 | 0.0124 | 0.0055 | 5.664 | 0.0126 | 60.4 | 87.7 | 97.2 | 98.1 | 100.0 | 100.0 |
MODWT-SVM1-c12 | 0.993 | 0.998 | 0.994 | 0.0248 | 0.0131 | 0.0064 | 9.493 | 0.0133 | 64.2 | 84.0 | 96.2 | 99.1 | 100.0 | 100.0 |
MODWT-SVM1-s18 | 0.996 | 0.999 | 0.996 | 0.0201 | 0.0155 | 0.0042 | 0.570 | 0.0158 | 38.7 | 68.9 | 98.1 | 100.0 | 100.0 | 100.0 |
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Seo, Y.; Choi, Y.; Choi, J. River Stage Modeling by Combining Maximal Overlap Discrete Wavelet Transform, Support Vector Machines and Genetic Algorithm. Water 2017, 9, 525. https://doi.org/10.3390/w9070525
Seo Y, Choi Y, Choi J. River Stage Modeling by Combining Maximal Overlap Discrete Wavelet Transform, Support Vector Machines and Genetic Algorithm. Water. 2017; 9(7):525. https://doi.org/10.3390/w9070525
Chicago/Turabian StyleSeo, Youngmin, Yunyoung Choi, and Jeongwoo Choi. 2017. "River Stage Modeling by Combining Maximal Overlap Discrete Wavelet Transform, Support Vector Machines and Genetic Algorithm" Water 9, no. 7: 525. https://doi.org/10.3390/w9070525