Forecasting Daily Solar Radiation Using CEEMDAN Decomposition-Based MARS Model Trained by Crow Search Algorithm
Abstract
:1. Introduction
2. Data Collection
3. Methodology of CEEMDAN and Data-Driven Models
3.1. Complete Ensemble EMD with Adaptive Noise (CEEMDAN)
3.2. Gene Expression Programing (GEP)
3.3. Multivariate Adaptive Regression Splines (MARS)
3.4. Crow Search Algorithm (CSA)
- Defining the optimization problem with all of its constraints, determining the decision variables and setting the CSA parameters, flock size (N), the flight length (fl), maximizing the iteration number (itermax), and the awareness probability (AP).
- Initializing the position and memory in a d-dimensional search space randomly for crows according to Equations (11) and (12). Each crow can be suitable solution for the problem and d indicates the quantity of the decision variables.
- Evaluating the fitness function for each crow by placing the decision variables into the objective function.
- Generating new positions as follows: crow i can generate a new situation and select ones among other cows (crow j) randomly and follows it to discover crow j’s hidden food source.
- The possibility of the new positions for all crows is checked as follows: If new position of crows is possible, the position of that crow is updated. Else, the crow remains in the current situation and new position is not generated for that crow.
- The fitness function for the new position of each crow is evaluated.
- The crows update their memory by Equation (13):
3.5. Self-Adaptive Multivariate Adaptive Regression Splines (SaMARS)
4. Model Performance
- Root mean square error (RMSE)
- Relative root mean square error (RRMSE)
- Mean absolute error (MAE)
- Nash-Sutcliffe efficiency (NSE)
- Willmott’s index of agreement (WI)
- Legates-McCabe’s index (LMI)
5. Results and Discussion
5.1. Implementation of CEEMDAN Based Models
5.2. Standalone and Hybrid GEP Models
5.3. Standalone and Hybrid MARS Models
5.4. Standalone and Hybrid SaMARS Models
5.5. Comparison of Proposed Hybrid and Standalone Models
6. Conclusions and Future Research
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Statistical Parameter | Meteorological Variable | ||||||
---|---|---|---|---|---|---|---|
TA (°C) | RH (%) | VP (hPa) | SLP (hPa) | PE (cm) | SD (hr) | DSR (MJ/m2) | |
Busan station (calibration data, 2000–2012) | |||||||
Minimum | −7.2 | 11.3 | 0.8 | 992.2 | 0.2 | 0.0 | 0.0 |
Mean | 14.8 | 62.2 | 12.9 | 1015.5 | 3.1 | 6.1 | 14.1 |
Maximum | 30.1 | 99.0 | 37.2 | 1036.2 | 8.8 | 13.1 | 31.3 |
Standard deviation | 8.2 | 18.8 | 8.4 | 7.1 | 1.5 | 3.9 | 7.0 |
Coefficient of variation | 66.5 | 354.7 | 70.5 | 51.1 | 2.2 | 14.9 | 49.4 |
Skewness index | −0.3 | −0.2 | 0.5 | −0.1 | 0.5 | −0.3 | 0.1 |
Busan station (validation data, 2013–2016) | |||||||
Minimum | −7.6 | 17.6 | 1.1 | 992.7 | 0.0 | 0.0 | 0.2 |
Mean | 15.4 | 64.1 | 13.4 | 1015.8 | 3.4 | 7.1 | 14.0 |
Maximum | 31.7 | 99.9 | 33.1 | 1034.4 | 11.5 | 13.1 | 28.7 |
Standard deviation | 8.1 | 17.8 | 8.3 | 7.4 | 1.7 | 4.1 | 7.0 |
Coefficient of variation | 66.0 | 318.6 | 68.6 | 55.3 | 2.9 | 17.0 | 48.3 |
Skewness index | −0.3 | −0.2 | 0.4 | −0.1 | 0.5 | −0.6 | 0.1 |
Incheon station (calibration data, 2000–2012) | |||||||
Minimum | −14.6 | 25.0 | 0.9 | 990.8 | 0.0 | 0.0 | 0.0 |
Mean | 12.6 | 67.3 | 12.2 | 1016.1 | 3.0 | 6.0 | 12.9 |
Maximum | 31.1 | 100.0 | 34.1 | 1039.0 | 12.0 | 13.7 | 32.1 |
Standard deviation | 9.8 | 15.0 | 8.3 | 8.1 | 1.8 | 3.9 | 6.9 |
Coefficient of variation | 97.0 | 224.8 | 68.3 | 65.4 | 3.3 | 15.5 | 47.1 |
Skewness index | −0.3 | −0.1 | 0.6 | 0.0 | 0.7 | −0.2 | 0.2 |
Incheon station (validation data, 2013–2016) | |||||||
Minimum | −13.1 | 31.0 | 1.3 | 991.2 | 0.0 | 0.0 | 0.5 |
Mean | 12.8 | 77.3 | 14.2 | 1016.4 | 3.4 | 7.1 | 12.5 |
Maximum | 30.8 | 99.0 | 38.3 | 1037.6 | 10.1 | 13.9 | 26.0 |
Standard deviation | 10.1 | 14.8 | 9.6 | 8.3 | 2.0 | 3.9 | 6.2 |
Coefficient of variation | 101.9 | 217.6 | 92.2 | 69.3 | 4.1 | 15.5 | 39.0 |
Skewness index | −0.3 | −0.4 | 0.6 | 0.0 | 0.5 | −0.5 | 0.2 |
Model | Design Parameter | |||||||
---|---|---|---|---|---|---|---|---|
GEP | Chromosomes | Gene size | Head size | Linking function | Mutation rate | Crossover rate | One and two- point recombination rate | IS and RIS transposition rate |
30 | 3 | 8 | Addition | 0.01 | 0.8 | 0.3 | 0.1 | |
MARS | Max function | GCV | Self- interactions | Max interactions | Threshold | Prune | - | - |
25–40 | 0, 2–4 | No | 2–4 | 1.0 × 10−4 | Yes | - | - |
Model (Station) | RMSE (MJ/m2) | RRMSE | MAE (MJ/m2) | NSE | WI | LMI |
---|---|---|---|---|---|---|
Calibration phase | ||||||
GEP (Busan) | 2.395 | 0.169 | 1.904 | 0.883 | 0.968 | 0.674 |
CEEMDAN-GEP (Busan) | 2.173 | 0.141 | 1.727 | 0.905 | 0.978 | 0.712 |
GEP (Incheon) | 2.408 | 0.193 | 1.872 | 0.877 | 0.968 | 0.675 |
CEEMDAN-GEP (Incheon) | 2.199 | 0.171 | 1.693 | 0.897 | 0.972 | 0.706 |
Validation phase | ||||||
GEP (Busan) | 2.992 | 0.189 | 2.402 | 0.814 | 0.955 | 0.585 |
CEEMDAN-GEP (Busan) | 2.514 | 0.171 | 1.995 | 0.873 | 0.974 | 0.654 |
GEP (Incheon) | 3.086 | 0.211 | 2.501 | 0.755 | 0.944 | 0.528 |
CEEMDAN-GEP (Incheon) | 2.862 | 0.192 | 2.324 | 0.789 | 0.952 | 0.564 |
Model (Station) | RMSE (MJ/m2) | RRMSE | MAE (MJ/m2) | NSE | WI | LMI |
---|---|---|---|---|---|---|
Calibration phase | ||||||
MARS (Busan) | 2.742 | 0.194 | 2.183 | 0.847 | 0.957 | 0.626 |
CEEMDAN-MARS (Busan) | 2.205 | 0.156 | 1.762 | 0.901 | 0.973 | 0.698 |
MARS (Incheon) | 2.346 | 0.1824 | 1.832 | 0.883 | 0.968 | 0.681 |
CEEMDAN-MARS (Incheon) | 1.901 | 0.146 | 1.506 | 0.923 | 0.979 | 0.738 |
Validation phase | ||||||
MARS (Busan) | 2.951 | 0.207 | 2.437 | 0.819 | 0.951 | 0.581 |
CEEMDAN-MARS (Busan) | 2.412 | 0.166 | 1.983 | 0.879 | 0.969 | 0.667 |
MARS (Incheon) | 3.066 | 0.214 | 2.421 | 0.758 | 0.948 | 0.544 |
CEEMDAN-MARS (Incheon) | 2.659 | 0.185 | 2.221 | 0.818 | 0.957 | 0.582 |
Model (Station) | RMSE (MJ/m2) | RRMSE | MAE (MJ/m2) | NSE | WI | LMI |
---|---|---|---|---|---|---|
Calibration phase | ||||||
SaMARS (Busan) | 2.386 | 0.169 | 1.887 | 0.884 | 0.968 | 0.677 |
CEEMDAN-SaMARS (Busan) | 1.828 | 0.129 | 1.431 | 0.932 | 0.982 | 0.776 |
SaMARS (Incheon) | 2.198 | 0.171 | 0.1714 | 0.897 | 0.972 | 0.702 |
CEEMDAN-SaMARS (Incheon) | 1.001 | 0.077 | 0.615 | 0.978 | 0.994 | 0.893 |
Validation phase | ||||||
SaMARS (Busan) | 2.562 | 0.174 | 2.092 | 0.864 | 0.964 | 0.638 |
CEEMDAN-SaMARS (Busan) | 2.427 | 0.181 | 1.894 | 0.878 | 0.971 | 0.672 |
SaMARS (Incheon) | 2.999 | 0.206 | 2.455 | 0.769 | 0.948 | 0.537 |
CEEMDAN-SaMARS (Incheon) | 2.311 | 0.164 | 1.931 | 0.883 | 0.967 | 0.659 |
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Rezaie-Balf, M.; Maleki, N.; Kim, S.; Ashrafian, A.; Babaie-Miri, F.; Kim, N.W.; Chung, I.-M.; Alaghmand, S. Forecasting Daily Solar Radiation Using CEEMDAN Decomposition-Based MARS Model Trained by Crow Search Algorithm. Energies 2019, 12, 1416. https://doi.org/10.3390/en12081416
Rezaie-Balf M, Maleki N, Kim S, Ashrafian A, Babaie-Miri F, Kim NW, Chung I-M, Alaghmand S. Forecasting Daily Solar Radiation Using CEEMDAN Decomposition-Based MARS Model Trained by Crow Search Algorithm. Energies. 2019; 12(8):1416. https://doi.org/10.3390/en12081416
Chicago/Turabian StyleRezaie-Balf, Mohammad, Niloofar Maleki, Sungwon Kim, Ali Ashrafian, Fatemeh Babaie-Miri, Nam Won Kim, Il-Moon Chung, and Sina Alaghmand. 2019. "Forecasting Daily Solar Radiation Using CEEMDAN Decomposition-Based MARS Model Trained by Crow Search Algorithm" Energies 12, no. 8: 1416. https://doi.org/10.3390/en12081416