The Ångström–Prescott Regression Coefficients for Six Climatic Zones in South Africa
Abstract
:1. Introduction
2. Materials and Methods
- Mean Bias Error (MBE), which estimates the average error in the prediction. A positive MBE indicates that the prediction is overestimated and vice versa; the lower values of MBE indicate a strong correlation between the prediction and observation. A relative Mean Bias Error (rMBE), which measures the size of the error in percentage terms, was also calculated. The metrices are expressed as:
- Mean Absolute Error (MAE), which measures the absolute value of the differences between the observed and the predicted values, gives a better idea of the prediction accuracy; relative Mean Absolute Error (rMAE), which measures the size of the error in percentage terms, was also calculated. The caution with MBE and rMBE is with the cancelling of positive and negative bias, which can lead to a false interpretation. The metrics are expressed as:
- Root Mean Square Error (RMSE), which compares the predicted and observed datasets, measures the statistical variability of the prediction accuracy and is expressed as shown in Equation (14), while Equation (15) shows the relative Root Mean Square Error (rRMSE), which measures the size error in percentage terms. The RMSE and rRMSE are also indifferent to the direction of the error. They are considered in this study since these put extra weight on large errors. The metrices are expressed as:
- Coefficient of Determination (R2), which is a statistical measure of the strength of the relationship between the movement of predicted and observed. R2 also measures how well the regression line represents the data. The value of R2 is such that . The closer R2 is to 1, the better the prediction. The metric is expressed as:
3. Results and Discussion
3.1. Annual AP Results
3.2. Validation Results
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Abbreviation | Full Description | Units |
SAWS | South African Weather Service | |
AP | Ångström–Prescott | |
a | Ångström–Prescott regression coefficient | |
b | Ångström–Prescott regression coefficient | |
GHI | Global Horizontal Irradiance | W/m2 |
Daily extraterrestrial or Top of the atmosphere global horizontal irradiance | W/m2 | |
TOA | Top of the Atmosphere | W/m2 |
N | Daily astronomical day length | Hours |
n | Daily measured sunshine duration | Hours |
BSRN | Baseline Solar Radiation Network | |
QC | Quality control | |
SPA | Solar Position Algorithm | |
Solar constant | W/m2 | |
Eccentricity factor | Degrees | |
Sunset hour angle | Degrees | |
Degrees of latitude | Degrees | |
δ | Solar declination | Degrees |
MBE | Mean Bias Error | MJ m−2d−1 |
rMBE | relative Mean Bias Error | Percentage (%) |
MAE | Mean Absolute Error | MJ m−2d−1 |
rMAE | relative Mean Absolute Error | Percentage (%) |
RMSE | Root Mean Square Error | MJ m−2d−1 |
rRMSE | relative Root Mean Square Error | Percentage (%) |
R2 | Correlation coefficient | |
NaN | Not a Number | |
D | Day of the year | |
CMP11 | Secondary standard Kipp & Zonen pyranometers |
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Station | Latitude (°) | Longitude (°) | Altitude (m) | Period | Climatic Zone |
---|---|---|---|---|---|
Upington | −28.48 | 21.12 | 848 | 2014-02-01 to 2019-11-30 | Arid Interior |
De Aar | −30.67 | 23.99 | 1284 | 2014-05-01 to 2019-12-31 | Cold Interior |
Irene | −25.91 | 28.21 | 1524 | 2014-03-01 to 2019-12-31 | Temperate Interior |
Mthatha | −31.55 | 28.67 | 744 | 2014-07-01 to 2019-12-31 | Subtropical Coastal |
George | −34.01 | 22.38 | 192 | 2015-01-01 to 2019-12-31 | Temperate Coastal |
Durban | −29.61 | 31.11 | 91 | 2015-03-01 to 2019-12-31 | Subtropical Coastal |
Polokwane | −23.86 | 29.45 | 1233 | 2015-03-01 to 2019-12-31 | Temperate Interior |
Thohoyandou | −23.08 | 30.38 | 619 | 2015-03-01 to 2017-10-31 | Hot Interior |
Station | a | b | RMBE | rMBE (%) | MAE | rMAE (%) | RMSE | rRMSE (%) | R2 |
---|---|---|---|---|---|---|---|---|---|
Upington | 0.243 | 0.549 | −0.360 | −0.120 | 0.841 | 0.311 | 1.061 | 0.393 | 0.930 |
De Aar | 0.191 | 0.600 | 0.733 | 0.371 | 1.136 | 0.506 | 1.375 | 0.598 | 0.930 |
Irene | 0.224 | 0.546 | 0.689 | 0.353 | 1.328 | 0.608 | 1.618 | 0.729 | 0.912 |
Mthatha | 0.210 | 0.562 | −0.104 | −0.013 | 1.168 | 0.582 | 1.474 | 0.735 | 0.951 |
George | 0.215 | 0.560 | −0.270 | −0.036 | 1.261 | 0.636 | 1.520 | 0.769 | 0.948 |
Durban | 0.207 | 0.540 | −0.322 | −0.106 | 1.425 | 0.745 | 1.741 | 0.910 | 0.915 |
Polokwane | 0.243 | 0.515 | −0.286 | −0.085 | 1.272 | 0.488 | 1.572 | 0.606 | 0.910 |
Thohoyandou | 0.188 | 0.571 | 0.286 | 0.168 | 1.071 | 0.550 | 1.433 | 0.746 | 0.937 |
Almorox et al. | 0.287 | 0.452 | −0.002 | - | - | - | 1.260 | - | - |
De Medeiros et al. | 0.39 | 0.29 | 1.040 | 6.29 | - | - | 1.94 | - | 0.58 |
Tsung et al. | 0.5 | 0.11 | 0.85 | 3.4 | 1.8 | - | 1.9 | - | - |
Zhang et al. | 0.214 | 0.552 | - | - | 2.249 | - | 0.214 | - | 0.875 |
Adamala et al. | 0.28 | 0.52 | - | - | - | - | 7.04 | - | 0.74 |
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Mabasa, B.; Lysko, M.D.; Tazvinga, H.; Mulaudzi, S.T.; Zwane, N.; Moloi, S.J. The Ångström–Prescott Regression Coefficients for Six Climatic Zones in South Africa. Energies 2020, 13, 5418. https://doi.org/10.3390/en13205418
Mabasa B, Lysko MD, Tazvinga H, Mulaudzi ST, Zwane N, Moloi SJ. The Ångström–Prescott Regression Coefficients for Six Climatic Zones in South Africa. Energies. 2020; 13(20):5418. https://doi.org/10.3390/en13205418
Chicago/Turabian StyleMabasa, Brighton, Meena D. Lysko, Henerica Tazvinga, Sophie T. Mulaudzi, Nosipho Zwane, and Sabata J. Moloi. 2020. "The Ångström–Prescott Regression Coefficients for Six Climatic Zones in South Africa" Energies 13, no. 20: 5418. https://doi.org/10.3390/en13205418