# A Quantile Functions-Based Investigation on the Characteristics of Southern African Solar Irradiation Data

^{1}

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## 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

**Theorem 1.**

- the uniform transformation rule applies and
- ordered U
_{r}leads to the corresponding ordered X_{r}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

_{(1)}, X

_{(2)}, X

_{(3)}, …, X

_{(n),}then any quantile distribution X = Q(p) can be generated from a uniform distribution U on the domain (0, 1) by X = Q(U). That is, ordering X corresponds to ordering U as in (5) here under:

_{(r)}, technically called the median rankit is defined as

#### 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

- y
_{sunrise}= y_{sunset}= 0. - y
_{sunrise−1hr}= y_{sunset+1hr}= 0.

#### 3.2.1. Venda and Gaborone Hourly Quantile Profiles

#### 3.2.2. Durban, Pretoria, Cape Town and Windhoek Hourly Quantile Profiles

^{−16}and 1.1102 × 10

^{−16}and standard deviations of 11.0653 and 13.4113 respectively. The residuals had also respective skewness of 0.051 and −0.055. As a result, the fitted QDFM is

_{NUST}= 0.2567696, µ

_{UP}= −1.15597) and standard deviation of (σ

_{NUST}= 21.3035529, σ

_{UP}= 2.77733). However, the residuals from the Windhoek and Pretoria deterministic models have respective skewness of 0.162308 and −0.1442648, which cannot be ignored (that is, the skewness cannot be approximated to zero). That is, the residuals are suggesting some skewness, so considering a skewed lambda quantile distribution (in Equation (16)) for the residuals will give better results [21]. Therefore, we fit the following QDFM for the Pretoria and Windhoek hourly profiles. Thus, the estimated parameters are shown in Table 6.

#### 3.2.3. Hourly Population Means

^{−1}(p). We adopt the method suggested by [26] of probabilistic polynomial approximations to evaluate the inverse. Researchers like [27,28] and the latest [29] concentrated on approximating the CDF. Ref. [29] are claiming to have the most accurate approximation using both the MATLAB Global Optimization Toolbox and BARON, but they did not document evaluating the inverse of the CDF. The approximation developed by [26] is explicit and has an acceptable maximum absolute percentage relative error (APRE) of 1.4 × 10

^{−2}. We find their approximation function simple and very accurate for the purposes of estimating the population mean SI in any time interval of interest. Therefore, Table 7 shows the estimated population mean of the average SI for 12:00, 13:00 and 14:00 time hours at each location.

^{2}which is the amount of energy required to fully charge a 12 Volt and 250 Amp solar battery. This means that given the correct solar panel capacity such a solar battery can be fully charged in at least five hours i.e., a period from 11:00 up to 15:00 at any of the locations in the Southern Africa region.

#### 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

**Figure A1.**Fitted residual distribution plot for (

**a**) Venda; (

**b**) Pretoria; (

**c**) Durban; (

**d**) Cape Town; (

**e**) Windhoek; (

**f**) Gaborone.

#### Appendix A.2. Hourly Profile QDFM Validation Plots

**Figure A2.**Fit-observation plot for (

**a**) Venda; (

**b**) Pretoria; (

**c**) Durban; (

**d**) Cape Town; (

**e**) Windhoek; (

**f**) Gaborone.

**Figure A3.**Distributional residual plots (

**a**) Venda; (

**b**) Pretoria; (

**c**) Durban; (

**d**) Cape Town; (

**e**) Windhoek; (

**f**) Gaborone.

## Appendix B

#### Appendix B.1. Fitted Probability Distributions on Modelling Residuals from Trigonometric Regression of Monthly Totals

**Figure A4.**Fitted residual distribution plot for (

**a**) Venda; (

**b**) Pretoria; (

**c**) Durban; (

**d**) Cape Town; (

**e**) Windhoek; (

**f**) Gaborone.

#### Appendix B.2. Monthly Total Profile QDFMS Validation Plots

**Figure A5.**Fit-observation plots (

**a**) Venda; (

**b**) Pretoria; (

**c**) Durban; (

**d**) Cape Town; (

**e**) Windhoek; (

**f**) Gaborone.

**Figure A6.**Distributional residual plots (

**a**) Venda; (

**b**) Pretoria; (

**c**) Durban; (

**d**) Cape Town; (

**e**) Windhoek; (

**f**) Gaborone.

## References

- Parzen, E. Quantile probability and statistical modelling. Stat. Sci.
**2004**, 19, 652–662. [Google Scholar] [CrossRef] - Gilchrist, W.G. Regression Revisited. Int. Stat. Rev.
**2008**, 76, 401–439. [Google Scholar] [CrossRef] - Yang, D. A universal benchmarking method for probabilistic solar irradiance forecasting. Sol. Energy
**2019**, 184, 410–416. [Google Scholar] [CrossRef] - Jain, P.K.; Lungu, E.M.; Prakash, J. Stochastic characteristics of solar irradiation—Extremum temperatures processes. In Proceedings of the World Renewable Energy Congress VII (WREC 2002), Cologne, Germany, 29 June–5 July 2002. [Google Scholar]
- Jain, P.K.; Prakash, J.; Lungu, E.M. Correlation between temperature and solar irradiation in Botswana: Bivariate model. In Proceedings of the 2nd IASTED Africa Conference Modelling and Simulation (Africa MS 2008), Gaborone, Botswana, 8–10 September 2008. [Google Scholar]
- Salima, G.; Chavuka, G.M.S. Determining Angstrom constants for estimating solar radiation in Malawi. Int. J. Geosci.
**2012**, 3, 391–397. [Google Scholar] [CrossRef] [Green Version] - Sivhugwana, K.S.; Ranganai, E. Intelligent techniques, harmonically coupled and SARIMA models in forecasting solar radiation data: A hybridisation approach. J. Energy South. Afr.
**2020**, 31, 14–37. [Google Scholar] [CrossRef] - Mutavhatsindi, T.; Sigauke, C.; Mbuvha, R. Forecasting Hourly Global Horizontal Solar Irradiance in South Africa. IEEE Access
**2020**, 8, 19887. [Google Scholar] [CrossRef] - Jain, P.K.; Lungu, E.M. Stochastic models for sunshine duration and solar irradiation. Renew. Energy
**2002**, 27, 197–209. [Google Scholar] [CrossRef] - Jain, P.K.; Prakash, J.; Lungu, E.M. Climate characteristics of Botswana. In Proceedings of the Sixth IASTED International Conference, Gaborone, Botswana, 11–13 September 2006. [Google Scholar]
- Madhlopa, A. Study of diurnal production of distilled water by using solar irradiation distribution about solar noon. In Proceedings of the EuroSun 2006 Conference, Glasgow, Scotland, 27–30 June 2006. [Google Scholar]
- Madhlopa, A. Solar radiation climate in Malawi. Sol. Energy
**2006**, 80, 1055–1057. [Google Scholar] [CrossRef] - Jain, P.K.; Lungu, E.M.; Prakash, J. Bivariate models: Relationships between solar irradiation and either sunshine or extremum temperatures. Renew. Energy
**2003**, 28, 1211–1223. [Google Scholar] [CrossRef] - Govender, P.; Brooks, M.J.; Mathews, A.P. Cluster analysis for classification and forecasting of solar irradiance in Durban, South Africa. J. Energy South. Afr.
**2018**, 29, 1–6. [Google Scholar] [CrossRef] - Bessafi, M.; Delage, O.; Jeanty, P.; Heintz, A.; Cazal, J.-D.; Delsaut, M.; Gangat, Y.; Partal, L.; Lan-Sun-Luk, J.-D.; Chabriat, J.-P.; et al. Research collaboration in solar radiometry between the University of Reunion Island and the University of Kwazulu-Natal. In Proceedings of the Third Southern African Solar Energy Conference, Mpumalanga, South Africa, 11–13 May 2015. [Google Scholar]
- Mpfumali, P.; Sigauke, C.; Bere, A.; Mlaudzi, S. Day Ahead Hourly Global Horizontal Irradiance Forecasting-Application to South African Data. Energies
**2019**, 12, 3569. [Google Scholar] [CrossRef] [Green Version] - Ranganai, E.; Sigauke, C. Capturing Long-Range Dependence and Harmonic Phenomena in 24-Hour Solar Irradiance Forecasting. IEEE Access
**2020**, 8, 172204–172218. [Google Scholar] [CrossRef] - Ratshilengo, M.; Sigauke, C.; Bere, A. Short-Term Solar Power Forecasting Using Genetic Algorithms: An Application Using South African Data. Appl. Sci.
**2021**, 11, 4214. [Google Scholar] [CrossRef] - Chandiwana, E.; Sigauke, C.; Bere, A. Twenty-four-hour ahead probabilistic global horizontal irradiation forecasting using Gaussian process regression. Algorithms
**2021**, 14, 177. [Google Scholar] [CrossRef] - Conde-Amboage, M.; Gonzalez-Manteiga, W.; Sanchez-Sellero, C. Quantile regression: Estimation and lack-of-fit tests. Bol. De Estad. E Investig. Oper.
**2018**, 34, 97–116. [Google Scholar] - Gilchrist, W.G. Statistical Modelling with Quantile Functions; Chapman and Hall/CRC: Boca Raton, FL, USA, 2007. [Google Scholar]
- Karian, Z.A.; Dudewicz, E.J. Handbook of Fitting Statistical Distributions with R.; Chapman and Hall/CRC: Boca Raton, FL, USA, 2010. [Google Scholar]
- Boland, J. Time series modelling of solar radiation. In Modelling Solar Radiation at the Earth’s Surface: Recent Advances; Badescu, V., Ed.; Springer-Verlag: Berlin/Heidelberg, Germany, 2008; Chapter 11; pp. 283–312. [Google Scholar]
- Delignette-Muller, M.-L.; Dutang, C.; Pouillot, R.; Denis, J.-B.; Siberchicot, A. Package ‘fitdistrplus’. J. Stat. Softw.
**2015**, 24, 1–14. [Google Scholar] - Stasinopoulos, D.M.; Rigby, A. Generalized additive models for location scale and shape (GAMLSS) in R. J. Stat. Softw.
**2007**, 23, 507–554. [Google Scholar] [CrossRef] [Green Version] - Richards, W.A.; Antoine, R.; Sahai, A.; Acharya, M.R. An Efficient Polynomial Approximation to the Normal Distribution Function and Its Inverse Function. J. Math. Res.
**2010**, 2, 47–51. [Google Scholar] [CrossRef] [Green Version] - Aludaat, K.M.; Alodat, M.T. A note on approximating the normal distribution function. Appl. Math. Sci.
**2008**, 2, 425–429. [Google Scholar] - Soranzo, A.; Epure, E. Very Simply Explicitly Invertible Approximations of Normal Cumulative and Normal Quantile Function. Appl. Math. Sci.
**2014**, 8, 4323–4341. [Google Scholar] [CrossRef] - Lipoth, J.; Tereda, Y.; Papalexiou, S.N.; Spiteri, R.J. A new very simply explicitly invertible approximation for the standard normal cumulative distribution function. AIMS Math.
**2022**, 7, 11635–11646. [Google Scholar] [CrossRef] - Yan, K.; Shen, H.; Wang, L.; Zhou, H.; Xu, M.; Mo, Y. Short-Term Solar Irradiance Forecasting Based on a Hybrid Deep Learning Methodology. Information
**2020**, 11, 32. [Google Scholar] [CrossRef] [Green Version] - Crowley, T.J. Causes of Climate Change Over the Past 1000 Years. Science
**2000**, 289, 270–277. [Google Scholar] [CrossRef] [Green Version] - Argueso, D.; Evans, J.P.; Fita, L.; Kathryn, J. Temperature response to future urbanization and climate change. Clim. Dyn.
**2014**, 42, 2183–2199. [Google Scholar] [CrossRef] - Chapman, S.; Watson, J.E.M.; Salazar, A.; Thatcher, M.; McAlpine, C.A. The impact of urbanization and climate change on urban temperatures: A systematic review. Landsc. Ecol.
**2017**, 32, 1921–1935. [Google Scholar] [CrossRef] - Paulescu, M.; Tulcan-Paulescu, E.; Sudhansu, S.S. A temperature-based model for global solar irradiance and its application to estimate daily irradiation values. Int. J. Energy Res.
**2011**, 35, 520–529. [Google Scholar] [CrossRef] - Mohanty, S.; Patra, P.K.; Sahoo, S.S. Prediction of global solar radiation using nonlinear autoregressive network with exogenous inputs (narx). In Proceedings of the 2015 39th National Systems Conference (NSC), IEEE, Greater Noida, India, 14–16 December 2015. [Google Scholar]
- Grantham, A.; Gel, Y.R.; Boland, J. Nonparametric short-term probabilistic forecasting for solar radiation. Sol. Energy
**2016**, 133, 465–475. [Google Scholar] [CrossRef] - Boland, J. Characterising seasonality of solar radiation and solar farm output. Energies
**2020**, 13, 471. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Radiometric Stations in Southern Africa (Source: www.sauran.ac.zw, accessed on 12 June 2022).

**Figure 3.**Monthly total solar irradiation for (

**a**) Venda; (

**b**) Pretoria; (

**c**) Durban; (

**d**) Cape Town; (

**e**) Windhoek; (

**f**) Gaborone.

Name of Plot | y | Against | Comment |
---|---|---|---|

Fit observation | x_{(}_{r}_{)} | Q’(p_{r}) | Points to exhibit an approximately linear pattern |

Distributional plots | f_{r} = x_{(}_{r}_{)} − Q’(p_{r}) | Q’(p_{r}) | 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 | ${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ${\widehat{\mathit{\beta}}}_{2}$ | ${\widehat{\mathit{\beta}}}_{3}$ | ${\widehat{\mathit{\beta}}}_{4}$ | ${\widehat{\mathit{\beta}}}_{5}$ | ${\widehat{\mathit{\beta}}}_{6}$ | ${\widehat{\mathit{\eta}}}_{}$ |
---|---|---|---|---|---|---|---|---|

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 | ${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ${\widehat{\mathit{\beta}}}_{2}$ | ${\widehat{\mathit{\beta}}}_{3}$ | ${\widehat{\mathit{\beta}}}_{4}$ | ${\widehat{\mathit{\beta}}}_{5}$ | ${\widehat{\mathit{\beta}}}_{6}$ | ${\widehat{\mathit{\eta}}}_{}$ |
---|---|---|---|---|---|---|---|---|

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 | $\mu +\sigma {\mathrm{\Phi}}^{-1}(p)$ |

Lognormal | Exp$(\mu +\sigma {\mathrm{\Phi}}^{-1}(p))$ |

Skewed Lambda | $\frac{1}{2\sigma}((1-\delta ){p}^{\sigma}-(1+\delta ){(1-p)}^{\sigma})$ |

Weibull | $\alpha {(-\mathrm{log}(1-p))}^{1/\gamma}$ |

Gumbel | $\alpha +\gamma \mathrm{log}{(-\mathrm{log}(1-p))}^{}$ |

Reverse Gumbel | $\alpha -\gamma \mathrm{log}{(-\mathrm{log}(1-p))}^{}$ |

Logistic | $\alpha +\gamma \mathrm{log}\left({\scriptscriptstyle \frac{p}{1-p}}\right)$ |

Cauchy | $\alpha +\gamma Tan{(\pi (p-0.5))}^{}$ |

Weibull Type 3 | $\beta {(-\mathrm{log}(1-p))}^{1/\gamma}$ |

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 | ${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ${\widehat{\mathit{\beta}}}_{2}$ | ${\widehat{\mathit{\eta}}}_{}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\gamma}}$ |
---|---|---|---|---|---|---|---|

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 | ${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ${\widehat{\mathit{\beta}}}_{2}$ | ${\widehat{\mathit{\eta}}}_{}$ | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ |

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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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Maposa, 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