A Stacked Machine Learning Algorithm for Multi-Step Ahead Prediction of Soil Moisture
Abstract
:1. Introduction
2. Materials and Methods
2.1. Standalone Machine Learning Algorithms
2.2. Evaluation Criteria
2.3. Stacked Model Development
Algorithm | Hyperparameter | Value |
---|---|---|
MLP | Number of hidden layers | 1 |
Number of hidden neurons | 5 | |
Activation function | Sigmoid | |
RF | Number of trees | 100 |
SVR | Kernel function | RBF |
C | 2 | |
ε | 0.01 | |
EN | α | 0.3 |
2.4. Case Study
3. Results
4. Discussion
- In the case of Model A, only the SVR-based variant was characterized by an appreciable bias, whereas in the case of Model B, an appreciable bias could be found in both the MLP- and SVR-based variants.
- The distribution of the relative error in both models was asymmetrical in many cases.
- The error distribution tended to become flatter as the forecast horizon increased, and the IQR of the relative error expanded as the forecast horizon increased. The standard deviation of the residuals increased as the forecasting horizon increased. For example, with reference to Model A based on the SM, it was 0.874, 1.049, and 1.164% for the 1-day-ahead, 2-days-ahead, and 3-days-ahead forecasts, respectively. With reference to the SM-based Model B, the standard deviation of the residuals was 0.978, 1.227, and 1.414 for the 1-day-ahead, 2-days-ahead, and 3-days-ahead forecasts, respectively.
- The number of outliers resulting from forecasting models was very low.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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SWC [%] | Air Temp. [°C] | Wind Speed [m/s] | Rel. Hum. [%] | |
---|---|---|---|---|
Mean | 24.18 | 11.06 | 3.28 | 80.12 |
Median | 25.00 | 11.12 | 3.03 | 81.43 |
Max | 34.20 | 27.36 | 8.52 | 99.62 |
Min | 9.40 | −4.82 | 0.67 | 53.36 |
St. Deviation | 5.16 | 5.54 | 1.42 | 9.50 |
CV | 0.21 | 0.50 | 0.43 | 0.12 |
1st Quartile | 20.55 | 6.79 | 2.20 | 73.00 |
3rd Quartile | 27.90 | 15.50 | 4.14 | 87.82 |
Skewness | −0.57 | 0.00 | 0.88 | −0.31 |
MLP | RF | SVR | Stacked Model | |||
---|---|---|---|---|---|---|
Model A (Training) | 1-day-ahead | R2 | 0.957 | 0.992 | 0.942 | 0.968 |
RMSE | 1.092 | 0.49 | 1.267 | 0.937 | ||
MAE | 0.816 | 0.356 | 0.911 | 0.694 | ||
MAPE | 3.36% | 1.49% | 3.73% | 2.85% | ||
2-days-ahead | R2 | 0.940 | 0.985 | 0.912 | 0.953 | |
RMSE | 1.285 | 0.663 | 1.569 | 1.137 | ||
MAE | 1.009 | 0.469 | 1.139 | 0.861 | ||
MAPE | 4.22% | 1.94% | 4.68% | 3.56% | ||
3-days-ahead | R2 | 0.928 | 0.977 | 0.891 | 0.941 | |
RMSE | 1.406 | 0.829 | 1.752 | 1.276 | ||
MAE | 1.101 | 0.571 | 1.266 | 0.959 | ||
MAPE | 4.66% | 2.36% | 5.24% | 3.99% | ||
Model A (Testing) | 1-day-ahead | R2 | 0.957 | 0.956 | 0.951 | 0.962 |
RMSE | 0.924 | 0.985 | 0.996 | 0.877 | ||
MAE | 0.741 | 0.787 | 0.744 | 0.673 | ||
MAPE | 3.41% | 3.62% | 3.35% | 3.05% | ||
2-days-ahead | R2 | 0.940 | 0.938 | 0.927 | 0.946 | |
RMSE | 1.146 | 1.217 | 1.264 | 1.053 | ||
MAE | 0.942 | 0.990 | 0.945 | 0.821 | ||
MAPE | 4.40% | 4.59% | 4.27% | 3.74% | ||
3-days-ahead | R2 | 0.921 | 0.929 | 0.911 | 0.935 | |
RMSE | 1.355 | 1.360 | 1.442 | 1.169 | ||
MAE | 1.105 | 1.113 | 1.069 | 0.921 | ||
MAPE | 5.25% | 5.22% | 4.83% | 4.22% |
MLP | RF | SVR | Stacked Model | |||
---|---|---|---|---|---|---|
Model B (Training) | 1-day-ahead | R2 | 0.946 | 0.990 | 0.934 | 0.965 |
RMSE | 1.222 | 0.533 | 1.365 | 0.989 | ||
MAE | 0.914 | 0.394 | 0.979 | 0.737 | ||
MAPE | 3.72% | 1.62% | 3.94% | 3.02% | ||
2-days-ahead | R2 | 0.919 | 0.976 | 0.892 | 0.943 | |
RMSE | 1.495 | 0.835 | 1.749 | 1.258 | ||
MAE | 1.161 | 0.586 | 1.274 | 0.964 | ||
MAPE | 4.77% | 2.38% | 5.13% | 3.98% | ||
3-days-ahead | R2 | 0.900 | 0.960 | 0.863 | 0.925 | |
RMSE | 1.658 | 1.073 | 1.989 | 1.441 | ||
MAE | 1.286 | 0.745 | 1.479 | 1.109 | ||
MAPE | 5.32% | 3.01% | 5.98% | 4.62% | ||
Model B (Testing) | 1-day-ahead | R2 | 0.951 | 0.943 | 0.941 | 0.949 |
RMSE | 0.982 | 1.145 | 1.069 | 0.976 | ||
MAE | 0.745 | 0.937 | 0.810 | 0.751 | ||
MAPE | 3.42% | 4.28% | 3.64% | 3.39% | ||
2-days-ahead | R2 | 0.928 | 0.916 | 0.907 | 0.924 | |
RMSE | 1.249 | 1.456 | 1.381 | 1.224 | ||
MAE | 0.964 | 1.198 | 1.028 | 0.973 | ||
MAPE | 4.48% | 5.53% | 4.59% | 4.45% | ||
3-days-ahead | R2 | 0.903 | 0.896 | 0.880 | 0.902 | |
RMSE | 1.513 | 1.667 | 1.606 | 1.411 | ||
MAE | 1.185 | 1.381 | 1.193 | 1.144 | ||
MAPE | 5.56% | 6.43% | 5.32% | 5.29% |
ARIMAX | ||
---|---|---|
1-day-ahead | R2 | 0.944 |
RMSE | 1.046 | |
MAE | 0.831 | |
MAPE | 3.88% | |
2-days-ahead | R2 | 0.924 |
RMSE | 1.284 | |
MAE | 0.997 | |
MAPE | 5.23% | |
3-days-ahead | R2 | 0.910 |
RMSE | 1.542 | |
MAE | 1.124 | |
MAPE | 5.63% |
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Granata, F.; Di Nunno, F.; Najafzadeh, M.; Demir, I. A Stacked Machine Learning Algorithm for Multi-Step Ahead Prediction of Soil Moisture. Hydrology 2023, 10, 1. https://doi.org/10.3390/hydrology10010001
Granata F, Di Nunno F, Najafzadeh M, Demir I. A Stacked Machine Learning Algorithm for Multi-Step Ahead Prediction of Soil Moisture. Hydrology. 2023; 10(1):1. https://doi.org/10.3390/hydrology10010001
Chicago/Turabian StyleGranata, Francesco, Fabio Di Nunno, Mohammad Najafzadeh, and Ibrahim Demir. 2023. "A Stacked Machine Learning Algorithm for Multi-Step Ahead Prediction of Soil Moisture" Hydrology 10, no. 1: 1. https://doi.org/10.3390/hydrology10010001