Bayesian Model Averaging: A Unique Model Enhancing Forecasting Accuracy for Daily Streamflow Based on Different Antecedent Time Series
Abstract
:1. Introduction
2. Methodology
2.1. Multivariate Adaptive Regression Spline (MARS)
2.2. M5 Model Tree (M5Tree)
2.3. Kernel Extreme Learning Machines (KELM)
2.4. Bayesian Model Averaging (BMA)
2.5. Assessment of Models Performance
2.5.1. Root Mean Square Error (RMSE)
2.5.2. Nash-Sutcliffe Efficiency (NSE)
2.5.3. Correlation Coefficient (R)
2.5.4. Mean Absolute Error (MAE)
3. Study Area and Data
4. Application and Results
4.1. Hongcheon Station
4.2. Jucheon Station
4.3. Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Hongcheon | Jucheon | |||
---|---|---|---|---|
Training | Testing | Training | Testing | |
Number | 2922 | 731 | 2922 | 731 |
Maximum | 1951.5 | 1362 | 2720.4 | 515.5 |
Minimum | 2.92 | 0.92 | 0.01 | 1.12 |
Average | 67.245 | 32.552 | 27.519 | 16.203 |
Standard Deviation | 111.949 | 83.321 | 105.677 | 39.099 |
Skewness | 4.844 | 9.310 | 11.966 | 7.207 |
Types | Input Combinations | Functions |
---|---|---|
M1 | t − 1 | Q(t) = f (Q(t − 1)) |
M2 | t − 1, t − 2 | Q(t) = f (Q(t − 1), Q(t − 2)) |
M3 | t − 1, t − 2, t − 3 | Q(t) = f (Q(t − 1), Q(t − 2), Q(t − 3)) |
M4 | t − 1, t − 3, t − 5 | Q(t) = f (Q(t − 1), Q(t − 3), Q(t − 5)) |
M5 | t − 2, t − 4, t − 6 | Q(t) = f (Q(t − 2), Q(t − 4), Q(t − 6)) |
M6 | t − 1, t − 2, t − 3, t − 4, t − 5, t − 6 | Q(t) = f (Q(t − 1), Q(t − 2), Q(t − 3), Q(t − 4), Q(t − 5), Q(t − 6)) |
Category | Assessment Indexes | MARS1 | M5Tree1 | KELM1 | BMA1 |
---|---|---|---|---|---|
RMSE (m3/s) | 52.214 | 52.866 | 51.541 | 50.887 | |
M1 | NSE | 0.609 | 0.600 | 0.619 | 0.629 |
R | 0.780 | 0.780 | 0.789 | 0.798 | |
MAE (m3/s) | 15.510 | 14.860 | 15.890 | 19.280 | |
MARS2 | M5Tree2 | KELM2 | BMA2 | ||
RMSE (m3/s) | 56.026 | 55.442 | 55.160 | 49.507 | |
M2 | NSE | 0.550 | 0.560 | 0.564 | 0.649 |
R | 0.743 | 0.749 | 0.751 | 0.812 | |
MAE (m3/s) | 16.210 | 15.160 | 14.740 | 17.200 | |
MARS3 | M5Tree3 | KELM3 | BMA3 | ||
RMSE (m3/s) | 54.280 | 57.704 | 51.242 | 50.212 | |
M3 | NSE | 0.578 | 0.523 | 0.624 | 0.639 |
R | 0.760 | 0.732 | 0.789 | 0.805 | |
MAE (m3/s) | 15.500 | 14.320 | 13.470 | 15.010 | |
MARS4 | M5Tree4 | KELM4 | BMA4 | ||
RMSE (m3/s) | 58.394 | 52.528 | 53.611 | 49.933 | |
M4 | NSE | 0.511 | 0.605 | 0.588 | 0.643 |
R | 0.715 | 0.779 | 0.767 | 0.808 | |
MAE (m3/s) | 16.110 | 15.720 | 14.080 | 14.720 | |
MARS5 | M5Tree5 | KELM5 | BMA5 | ||
RMSE (m3/s) | 71.612 | 72.264 | 69.496 | 69.283 | |
M5 | NSE | 0.266 | 0.252 | 0.308 | 0.313 |
R | 0.524 | 0.505 | 0.555 | 0.562 | |
MAE (m3/s) | 26.860 | 23.910 | 21.360 | 25.040 | |
MARS6 | M5Tree6 | KELM6 | BMA6 | ||
RMSE (m3/s) | 54.154 | 58.372 | 51.473 | 51.212 | |
M6 | NSE | 0.580 | 0.512 | 0.620 | 0.624 |
R | 0.762 | 0.715 | 0.794 | 0.790 | |
MAE (m3/s) | 15.570 | 16.310 | 15.690 | 14.410 |
Category | Assessment Indexes | MARS1 | M5Tree1 | KELM1 | BMA1 |
---|---|---|---|---|---|
RMSE (m3/s) | 30.429 | 31.367 | 29.498 | 28.396 | |
M1 | NSE | 0.397 | 0.360 | 0.434 | 0.475 |
R | 0.688 | 0.670 | 0.664 | 0.689 | |
MAE (m3/s) | 9.910 | 10.390 | 7.380 | 7.850 | |
MARS2 | M5Tree2 | KELM2 | BMA2 | ||
RMSE (m3/s) | 34.026 | 31.972 | 29.730 | 29.083 | |
M2 | NSE | 0.247 | 0.335 | 0.425 | 0.449 |
R | 0.642 | 0.654 | 0.667 | 0.670 | |
MAE (m3/s) | 10.760 | 9.900 | 7.600 | 7.840 | |
MARS3 | M5Tree3 | KELM3 | BMA3 | ||
RMSE (m3/s) | 32.925 | 32.554 | 30.346 | 29.321 | |
M3 | NSE | 0.294 | 0.310 | 0.401 | 0.440 |
R | 0.656 | 0.658 | 0.648 | 0.664 | |
MAE (m3/s) | 10.490 | 13.180 | 7.700 | 7.940 | |
MARS4 | M5Tree4 | KELM4 | BMA4 | ||
RMSE (m3/s) | 31.896 | 31.648 | 29.923 | 28.972 | |
M4 | NSE | 0.337 | 0.347 | 0.416 | 0.454 |
R | 0.654 | 0.665 | 0.651 | 0.674 | |
MAE (m3/s) | 10.540 | 11.720 | 7.550 | 7.990 | |
MARS5 | M5Tree5 | KELM5 | BMA5 | ||
RMSE (m3/s) | 37.917 | 38.960 | 35.990 | 34.657 | |
M5 | NSE | 0.063 | 0.011 | 0.156 | 0.219 |
R | 0.465 | 0.415 | 0.445 | 0.469 | |
MAE (m3/s) | 15.410 | 19.110 | 11.960 | 11.870 | |
MARS6 | M5Tree6 | KELM6 | BMA6 | ||
RMSE (m3/s) | 30.690 | 31.417 | 29.723 | 29.092 | |
M6 | NSE | 0.386 | 0.357 | 0.424 | 0.449 |
R | 0.676 | 0.651 | 0.656 | 0.670 | |
MAE (m3/s) | 9.730 | 11.800 | 7.610 | 8.190 |
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Kim, S.; Alizamir, M.; Kim, N.W.; Kisi, O. Bayesian Model Averaging: A Unique Model Enhancing Forecasting Accuracy for Daily Streamflow Based on Different Antecedent Time Series. Sustainability 2020, 12, 9720. https://doi.org/10.3390/su12229720
Kim S, Alizamir M, Kim NW, Kisi O. Bayesian Model Averaging: A Unique Model Enhancing Forecasting Accuracy for Daily Streamflow Based on Different Antecedent Time Series. Sustainability. 2020; 12(22):9720. https://doi.org/10.3390/su12229720
Chicago/Turabian StyleKim, Sungwon, Meysam Alizamir, Nam Won Kim, and Ozgur Kisi. 2020. "Bayesian Model Averaging: A Unique Model Enhancing Forecasting Accuracy for Daily Streamflow Based on Different Antecedent Time Series" Sustainability 12, no. 22: 9720. https://doi.org/10.3390/su12229720