A Quantile Functions-Based Investigation on the Characteristics of Southern African Solar Irradiation Data
Abstract
:1. Introduction
1.1. Rationale of the Study
1.2. Contribution of the Study
1.3. Review of Literature
2. Materials and Methods
2.1. Quantile Functions
- the uniform transformation rule applies and
- ordered Ur leads to the corresponding ordered Xr such that
- If X has a quantile distribution, R(p), on the positive axis, 0 ≤ x < 1, then the distribution −R(1 − p) is the quantile distribution that is its reflection in the axis at x = 0, called the reflected distribution on −1 < x ≤ 0.
- The reciprocal 1/X has the reciprocal distribution 1/R(1 − p) also on 0 ≤ x < 1.
2.2. Method of Percentiles
2.3. Parameter Estimation
2.4. Model Validation
2.4.1. Graphical Analysis
2.4.2. Chi-Square Goodness of Fit Test
3. Results and Discussions
3.1. Ground-Based Data
3.2. Hourly Solar Irradiance Distributional Modelling
- ysunrise = ysunset = 0.
- ysunrise−1hr = ysunset+1hr = 0.
3.2.1. Venda and Gaborone Hourly Quantile Profiles
3.2.2. Durban, Pretoria, Cape Town and Windhoek Hourly Quantile Profiles
3.2.3. Hourly Population Means
3.3. Daily Total SI Distributional Modelling
3.4. Monthly Total SI Distribution Modelling
3.5. Model Validations
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix A.1. Fitted Probability Distributions on Modelling Residuals from Trigonometric Regression of the Hourly Profiles
Appendix A.2. Hourly Profile QDFM Validation Plots
Appendix B
Appendix B.1. Fitted Probability Distributions on Modelling Residuals from Trigonometric Regression of Monthly Totals
Appendix B.2. Monthly Total Profile QDFMS Validation Plots
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Name of Plot | y | Against | Comment |
---|---|---|---|
Fit observation | x(r) | Q’(pr) | Points to exhibit an approximately linear pattern |
Distributional plots | fr = x(r) − Q’(pr) | Q’(pr) | Points to be randomly distributed |
Station | Latitude | Longitude | Location | Period |
---|---|---|---|---|
University of Venda (UV) | −23.13100052 | 30.42399979 | Venda | April 2015–April 2022 |
University of Pretoria (UP) | −25.75308037 | 28.22859001 | Pretoria | July 2017–June 2021 |
University of KwaZulu-Natal Howard College (UKZNH) | −29.87097931 | 30.97694969 | Durban | December 2015–September 2022 |
Stellenbosch University (SUN) | −33.92810059 | 18.86540031 | Cape Town | July 2017–June 2021 |
Namibian University of Science and Technology (NUST) | −22.56500053 | 17.07500076 | Windhoek | July 2017–June 2021 |
University of Gaborone (UG) | −24.6609993 | 25.93400002 | Gaborone | January 2015–November 2020 |
Location | Shape | Scale | Skewness |
---|---|---|---|
Venda | 22.676906 | −2.308079 | −5.612271 |
Gaborone | 23.233404 | 2.127659 | −1.204687 |
Location | ||||||||
---|---|---|---|---|---|---|---|---|
Venda | 143.24 | −327.52 | −55.60 | 148.90 | 57.37 | −17.33 | 18.81 | 2.02 |
Gaborone | 422.36 | −372.09 | −92.34 | 163.73 | 71.42 | −16.13 | −6.71 | −8.17 |
Location | Metric | Normal | Cauchy |
---|---|---|---|
Durban | AIC | 187.4920 | 199.3287 |
BIC | 189.8481 | 201.6848 | |
Cape Town | AIC | 196.7216 | 211.7815 |
BIC | 199.077 | 214.1376 | |
Windhoek | AIC | 218.9350 | 222.8473 |
BIC | 221.2911 | 225.2034 |
Location | ||||||||
---|---|---|---|---|---|---|---|---|
Durban | 186.88 | −300.05 | −27.46 | 145.53 | 28.01 | −26.56 | −11.13 | 1.089 |
Cape Town | 220.88 | −309.44 | −111.00 | 110.03 | 91.52 | −6.93 | −11.83 | 1.034 |
Windhoek | 267.82 | −400.60 | −137.34 | 159.85 | 114.20 | −27.07 | −29.39 | 3676.63 |
Pretoria | 247.62 | −362.25 | −54.47 | 163.33 | 51.28 | −23.25 | −10.66 | −312.92 |
Location | 12:00 | 13:00 | 14:00 |
---|---|---|---|
Venda | 704.5501 | 724.3324 | 664.2824 |
Pretoria | 792.3848 | 798.1858 | 720.3530 |
Durban | 653.7334 | 646.0031 | 566.3265 |
Cape Town | 647.2710 | 702.8115 | 690.4624 |
Windhoek | 856.5969 | 927.0284 | 892.8881 |
Gaborone | 789.5647 | 814.5785 | 756.4473 |
Probability Distribution | Quantile Function |
---|---|
Normal | |
Lognormal | Exp |
Skewed Lambda | |
Weibull | |
Gumbel | |
Reverse Gumbel | |
Logistic | |
Cauchy | |
Weibull Type 3 |
Month | Venda | Pretoria | Durban | Cape Town | Windhoek | Gaborone |
---|---|---|---|---|---|---|
January | 5808.48 | 6570.46 | 7419.84 | 8350.78 * | 7966.67 | 7045.33 |
February | 5118.63 | 5796.38 | 5569.62 | 7339.92 * | 6655.05 | 6741.43 |
March | 5328.46 | 5549.78 | 5727.71 | 5478.89 | 6969.69 * | 5847.43 |
April | 4218.16 | 4563.87 | 3869.33 | 4241.18 | 5855.68 * | 5143.91 |
May | 4189.18 | 4626.59 | 2832.39 | 3321.19 | 5183.17 * | 4593.42 |
June | 4207.39 | 4002.05 | 3543.30 | 2380.00 | 4946.30 * | 4292.30 |
July | 4463.09 | 4554.78 | 3146.75 | 3077.00 | 5109.11 * | 4522.42 |
August | 4338.57 | 5237.01 | 4393.84 | 3331.33 | 10,342.86 * | 3966.38 |
September | 5820.81 | 6381.69 | 4684.33 | 4937.00 | 10,678.41 * | 6310.75 |
October | 5441.11 | 6508.65 | 5773.34 | 7396.06 * | 7342.81 | 6881.60 |
November | 5992.28 | 7045.96 | 5197.02 | 7909.29 | 8022.61 * | 7370.91 |
December | 5786.87 | 7165.13 | 7118.95 | 8392.25 | 8799.95 * | 6856.38 |
Maximum | 5992.28 | 7165.13 | 7419.84 | 8350.78 | 10,678.41 | 7370.91 |
Minimum | 4189.79 | 4002.05 | 2832.39 | 2379.96 | 4946.30 | 4292.30 |
Location | With | Without |
---|---|---|
Venda | 266.7684 | 265.613 |
Pretoria | 256.3586 | 255.4424 |
Location | Metric | Normal | Cauchy |
---|---|---|---|
Cape Town | AIC | 187.4920 | 199.3287 |
BIC | 189.8481 | 201.6848 | |
Durban | AIC | 268.5895 | 271.3327 |
BIC | 269.5593 | 272.3025 |
Location | Probability Distribution | ||||||
---|---|---|---|---|---|---|---|
Venda | R. Gumbel | 1,678,882.00 | −8767.19 | 40,937.26 | 2013.06 | −768.98 | 9.11 |
Pretoria | R. Gumbel | 3,692,969.00 | −9175.68 | 20,756.98 | 4163.51 | −852.62 | 8.72 |
Windhoek | SN2 | −24,798,121 | −5434.35 | 36,610.50 | 2870.26 | 8700.05 | −0.69 |
Location | Probability Distribution | ||||||
Cape Town | Normal | 155,245.11 | 12,380.08 | 82,328.01 | −39.04 | −2.31 × 10−16 | 11.06526 |
Durban | Normal | 197,409.84 | 3445.95 | 37,525.34 | 2536.12 | 2488.44 | 9834.54 |
Gaborone | Normal | 148,521.33 | 22,150.61 | 41,991.97 | 2372.42 | 2863 | 23,670.46 |
Location | Hourly QDFM | Monthly QDFM | ||
---|---|---|---|---|
HL | Runs test | HL | Runs test | |
Venda | 1 | 0.09498 | 1 | 0.0154 |
Pretoria | 1 | 1 | 1 | 0.2259 |
Durban | 1 | 0.4038 | 1 | 0.5431 |
Cape Town | 1 | 0.4038 | 1 | 0.2154 |
Windhoek | 1 | 0.4038 | 1 | 0.2259 |
Gaborone | 1 | 0.2105 | 1 | 0.0154 |
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Maposa, D.; Masache, A.; Mdlongwa, P. A Quantile Functions-Based Investigation on the Characteristics of Southern African Solar Irradiation Data. Math. Comput. Appl. 2023, 28, 86. https://doi.org/10.3390/mca28040086
Maposa D, Masache A, Mdlongwa P. A Quantile Functions-Based Investigation on the Characteristics of Southern African Solar Irradiation Data. Mathematical and Computational Applications. 2023; 28(4):86. https://doi.org/10.3390/mca28040086
Chicago/Turabian StyleMaposa, Daniel, Amon Masache, and Precious Mdlongwa. 2023. "A Quantile Functions-Based Investigation on the Characteristics of Southern African Solar Irradiation Data" Mathematical and Computational Applications 28, no. 4: 86. https://doi.org/10.3390/mca28040086