Modeling Monthly Rainfall Data Using the Alpha Power Transformed X-Lindley Distribution in the Toba Lake Region

Mohamad Khoirun Najib, Sri Nurdiati, Elis Khatizah, Aulia Rizki Firdawanti, Hendri Irwandi, Mirza Farhan Azhari, David Vijanarco Martal, Nicholas Abisha

Abstract


Modeling rainfall is crucial for hydrological studies and climate adaptation, especially in regions with complex topography such as the Toba Lake area, North Sumatra. Classical probability distributions often struggle to represent skewness, heavy tails, and variability observed in tropical rainfall. This study explores APTXL distribution as a flexible two-parameter model. Through the alpha power transformation, APTXL extends the X-Lindley distribution by introducing an additional shape parameter, allowing better accommodation of asymmetrical and extreme values while maintaining analytical tractability. Statistical properties are derived, and parameters are estimated using maximum likelihood. The model is applied to a long-term dataset from 13 meteorological stations, covering 408 monthly observations per station. Comparative analysis against Gamma, Lognormal, and Generalized Extreme Value distributions using multiple goodness-of-fit criteria indicates that APTXL provides consistently improved performance. These results suggest APTXL as a practical tool for rainfall modeling and water-resource applications in climate-sensitive regions.

Keywords


Alpha Power Transformation; Goodness-of-Fit Criteria; Maximum Likelihood Estimation; Probabilistic Rainfall Modeling; X-Lindley Distribution

Full Text:

PDF

References


E. Deal, J. Braun, and G. Botter, “Understanding the Role of Rainfall and Hydrology in Determining Fluvial Erosion Efficiency,” Journal of Geophysical Research: Earth Surface, vol. 123, no. 4, pp. 744–778, 2018, doi: 10.1002/2017JF004393.

Z. Sokol, J. Szturc, J. Orellana-Alvear, J. Popová, A. Jurczyk, and R. Célleri, “The role of weather radar in rainfall estimation and its application in meteorological and hydrological modelling —A review,” Remote Sensing, vol. 13, no. 3, pp. 1–38, 2021, doi: 10.3390/rs13030351.

M. Dumont et al., “Assessing rainfall global products reliability for water resource management in a tropical volcanic mountainous catchment,” Journal of Hydrology: Regional Studies, vol. 40, p. 101037, 2022, doi: 10.1016/j.ejrh.2022.101037.

E. Bessah, E. A. Boakye, S. K. Agodzo, E. Nyadzi, I. Larbi, and A. Awotwi, “Increased seasonal rainfall in the twenty-first century over Ghana and its potential implications for agriculture productivity,” Environment, Development and Sustainability, vol. 23, no. 8, pp. 12342–12365, 2021, doi: 10.1007/s10668-020-01171-5.

S. Benziane, “Survey: Rainfall Prediction Precipitation, Review of Statistical Methods,” WSEAS Transactions on Systems, vol. 23, pp. 47–59, 2024, doi: 10.37394/23202.2024.23.5.

J. A. Marengo, P. I. Camarinha, L. M. Alves, F. Diniz, and R. A. Betts, “Extreme Rainfall and Hydro-Geo-Meteorological Disaster Risk in 1.5, 2.0, and 4.0°C Global Warming Scenarios: An Analysis for Brazil,” Frontiers in Climate, vol. 3, p. 610433, 2021, doi: 10.3389/fclim.2021.610433.

E. Bevacqua, G. Zappa, F. Lehner, and J. Zscheischler, “Precipitation trends determine future occurrences of compound hot–dry events,” Nature Climate Change, vol. 12, no. 4, pp. 350–355, 2022, doi: 10.1038/s41558-022-01309-5.

L. Gimeno et al., “Extreme precipitation events,” Wiley Interdisciplinary Reviews: Water, vol. 9, no. 6, p. e1611, 2022, doi: 10.1002/wat2.1611.

R. Satyaningsih, V. Jetten, J. Ettema, A. Sopaheluwakan, L. Lombardo, and D. E. Nuryanto, “Dynamic rainfall thresholds for landslide early warning in Progo Catchment, Java, Indonesia,” Natural Hazards, vol. 119, no. 3, pp. 2133–2158, 2023, doi: 10.1007/s11069-023-06208-2.

E. Yanfatriani et al., “Extreme Rainfall Trends and Hydrometeorological Disasters in Tropical Regions: Implications for Climate Resilience,” Emerging Science Journal, vol. 8, no. 5, pp. 1860–1874, 2024, doi: 10.28991/ESJ-2024-08-05-012.

C. A. Chesner, “The Toba Caldera Complex,” Quaternary International, vol. 258, pp. 5–18, 2012, doi: 10.1016/j.quaint.2011.09.025.

H. Irwandi, M. S. Rosid, and T. Mart, “The effects of ENSO, climate change and human activities on the water level of Lake Toba, Indonesia: a critical literature review,” Geoscience Letters, vol. 8, no. 1, p. 21, 2021, doi: 10.1186/s40562-021-00191-x.

E. Cristiano, M. C. Ten Veldhuis, and N. Van De Giesen, “Spatial and temporal variability of rainfall and their effects on hydrological response in urban areas - A review,” Hydrology and Earth System Sciences, vol. 21, no. 7, pp. 3859–3878, 2017, doi: 10.5194/hess-21-3859-2017.

T. Meema, Y. Tachikawa, Y. Ichikawa, and K. Yorozu, “Real-time optimization of a large-scale reservoir operation in Thailand using adaptive inflow prediction with medium-range ensemble precipitation forecasts,” Journal of Hydrology: Regional Studies, vol. 38, 2021, doi: 10.1016/j.ejrh.2021.100939.

V. Anupoju, B. P. Kambhammettu, and S. K. Regonda, “Role of Short-Term Weather Forecast Horizon in Irrigation Scheduling and Crop Water Productivity of Rice,” Journal of Water Resources Planning and Management, vol. 147, no. 8, 2021, doi: 10.1061/(asce)wr.1943-5452.0001406.

Y. Zhang and J. M. Swaminathan, “Improved crop productivity through optimized planting schedules,” Manufacturing and Service Operations Management, vol. 22, no. 6, pp. 1165–1180, 2020, doi: 10.1287/MSOM.2020.0941.

I. G. Prihanto et al., “A technology acceptance model of satellite-based hydrometeorological hazards early warning system in Indonesia: an-extended technology acceptance model,” Cogent Business and Management, vol. 11, no. 1, 2024, doi: 10.1080/23311975.2024.2374880.

A. Susandi et al., “Development of hydro-meteorological hazard early warning system in Indonesia,” Journal of Engineering and Technological Sciences, vol. 50, no. 4, pp. 461–478, 2018, doi: 10.5614/j.eng.technol.sci.2018.50.4.2.

L. Alfieri and J. Thielen, “A European precipitation index for extreme rain-storm and flash flood early warning,” Meteorological Applications, vol. 22, no. 1, pp. 3–13, 2015, doi: 10.1002/met.1328.

W. Yuan et al., “Study on the Early Warning for Flash Flood Based on Random Rainfall Pattern,” Water Resources Management, vol. 36, no. 5, pp. 1587–1609, 2022, doi: 10.1007/s11269-022-03106-3.

D. Li, Z. Liu, D. Wang, and X. Liu, “Rainfall temporal variability-oriented optimization of urban water resources allocation,” Journal of Hydrology: Regional Studies, vol. 61, 2025, doi: 10.1016/j.ejrh.2025.102694.

M. Ali, R. C. Deo, Y. Xiang, Y. Li, and Z. M. Yaseen, “Forecasting long-term precipitation for water resource management: a new multi-step data-intelligent modelling approach,” Hydrological Sciences Journal, vol. 65, no. 16, pp. 2693–2708, 2020, doi: 10.1080/02626667.2020.1808219.

P. de S. M. P. Ximenes, A. S. Alves da Silva, F. Ashkar, and T. Stosic, “Best-fit probability distribution models for monthly rainfall of Northeastern Brazil,” Water Science and Technology, vol. 84, no. 6, pp. 1541–1556, 2021, doi: 10.2166/wst.2021.304.

N. F. E. M. Johar, A. Senawi, and N. A. A. A. Ghani, “An assessment of rainfall distribution in Kuantan river basin using generalized extreme value distribution and Gamma distribution,” in AIP Conference Proceedings, 2024. doi: 10.1063/5.0192286.

G. Arriola, L. Villegas, J. Fernandez, J. Vallejos, and C. Idrogo, “Assessment of Parameters of the Generalized Extreme Value Distribution in Rainfall of the Peruvian North,” Revista Politecnica, vol. 52, no. 2, pp. 99–112, 2023, doi: 10.33333/rp.vol52n2.10.

B. Moccia, C. Mineo, E. Ridolfi, F. Russo, and F. Napolitano, “Probability distributions of daily rainfall extremes in Lazio and Sicily, Italy, and design rainfall inferences,” Journal of Hydrology: Regional Studies, vol. 33, p. 100771, 2021, doi: 10.1016/j.ejrh.2020.100771.

Y. Hundecha, M. Pahlow, and A. Schumann, “Modeling of daily precipitation at multiple locations using a mixture of distributions to characterize the extremes,” Water Resources Research, vol. 45, no. 12, 2009, doi: 10.1029/2008WR007453.

J. Y. Shin, T. Lee, and T. B. M. J. Ouarda, “Heterogeneous mixture distributions for modeling multisource extreme rainfalls,” Journal of Hydrometeorology, vol. 16, no. 6, pp. 2639–2657, 2015, doi: 10.1175/JHM-D-14-0130.1.

M. K. Najib, S. Nurdiati, and A. Sopaheluwakan, “Copula-based joint distribution analysis of the ENSO effect on the drought indicators over Borneo fire-prone areas,” Modeling Earth Systems and Environment, vol. 8, no. 2, pp. 2817–2826, 2022, doi: 10.1007/s40808-021-01267-5.

S. Liu and X. Dong, “The return period analysis of heavy rainfall disasters based on copula joint statistical modeling,” Geomatics, Natural Hazards and Risk, vol. 16, no. 1, 2025, doi: 10.1080/19475705.2025.2483799.

N. Alsadat, “A new extension of XLindley distribution with mathematical properties, estimation, and application on the rainfall data,” Heliyon, vol. 10, no. 19, 2024, doi: 10.1016/j.heliyon.2024.e38143.

S. Dey, I. Ghosh, and D. Kumar, “Alpha-Power Transformed Lindley Distribution: Properties and Associated Inference with Application to Earthquake Data,” Annals of Data Science, vol. 6, no. 4, pp. 623–650, 2019, doi: 10.1007/s40745-018-0163-2.

S. J. Dugasa, A. T. Goshu, and B. G. Arero, “Alpha Power Transformation of the Lindley Probability Distribution,” Journal of Probability and Statistics, vol. 2024, no. 1, 2024, doi: 10.1155/2024/9068114.

F. Y. Eissa and C. D. Sonar, “Alpha Power Transformed Extended power Lindley Distribution,” Journal of Statistical Theory and Applications, vol. 22, no. 1–2, pp. 1–18, 2023, doi: 10.1007/s44199-022-00051-3.

A. M. Gemeay et al., “The power new XLindley distribution: Statistical inference, fuzzy reliability, and applications,” Heliyon, vol. 10, no. 17, 2024, doi: 10.1016/j.heliyon.2024.e36594.

P. J. Northrop, “Stochastic Models of Rainfall,” Annual Review of Statistics and Its Application, vol. 11, no. 1, pp. 51–74, 2024, doi: 10.1146/annurev-statistics-040622-023838.

M. F. R. Gaona, K. Michaelides, and M. B. Singer, “STORM v.2: A simple, stochastic rainfall model for exploring the impacts of climate and climate change at and near the land surface in gauged watersheds,” Geoscientific Model Development, vol. 17, no. 13, pp. 5387–5412, 2024, doi: 10.5194/gmd-17-5387-2024.

T. Rasool, S. Sahoo, R. Das Bhowmik, and D. Nagesh Kumar, “Development of a stochastic rainfall generator to yield unprecedented rainfall events,” Journal of Hydrology, vol. 641, 2024, doi: 10.1016/j.jhydrol.2024.131809.

R. Montes-Pajuelo, Á. M. Rodríguez-Pérez, R. López, and C. A. Rodríguez, “Analysis of Probability Distributions for Modelling Extreme Rainfall Events and Detecting Climate Change: Insights from Mathematical and Statistical Methods,” Mathematics, vol. 12, no. 7, 2024, doi: 10.3390/math12071093.

R. L. T. Henriksen, J. B. Hubrechts, J. K. Møller, P. Knudsen, and J. W. Pedersen, “Large-scale rain gauge network optimization using a kriging emulator,” Journal of Hydrology, vol. 637, 2024, doi: 10.1016/j.jhydrol.2024.131360.

S. M. Tabatabaei, M. Dastourani, S. Eslamian, and M. Nazeri Tahroudi, “Ranking and optimizing the rain-gauge networks using the entropy–copula approach (Case study of the Siminehrood Basin, Iran),” Applied Water Science, vol. 12, no. 9, 2022, doi: 10.1007/s13201-022-01735-y.

A. H. Hussein and M. N. Kasim, “Utilizing statistical distribution tests to develop rainfall intensity–duration–frequency curves for enhanced hydrological analysis in Kirkuk city, Iraq,” Water Practice and Technology, vol. 19, no. 11, pp. 4378–4389, 2024, doi: 10.2166/wpt.2024.258.

A. Mahdavi and D. Kundu, “A new method for generating distributions with an application to exponential distribution,” Communications in Statistics - Theory and Methods, vol. 46, no. 13, pp. 6543–6557, 2017, doi: 10.1080/03610926.2015.1130839.

M. Nassar, A. Alzaatreh, M. Mead, and O. Abo-Kasem, “Alpha power Weibull distribution: Properties and applications,” Communications in Statistics - Theory and Methods, vol. 46, no. 20, pp. 10236–10252, 2017, doi: 10.1080/03610926.2016.1231816.

A. M. Basheer, “Alpha power inverse Weibull distribution with reliability application,” Journal of Taibah University for Science, vol. 13, no. 1, pp. 423–432, 2019, doi: 10.1080/16583655.2019.1588488.

S. Dey, M. Nassar, and D. Kumar, “Alpha power transformed inverse Lindley distribution: A distribution with an upside-down bathtub-shaped hazard function,” Journal of Computational and Applied Mathematics, vol. 348, pp. 130–145, 2019, doi: 10.1016/j.cam.2018.03.037.

S. Ihtisham, A. Khalil, S. Manzoor, S. A. Khan, and A. Ali, “Alpha-Power Pareto distribution: its properties and applications,” PLoS ONE, vol. 14, p. e0218027, 2019.

R. A. Zeineldin, M. Ahsan Ul Haq, S. Hashmi, and M. Elsehety, “Alpha Power Transformed Inverse Lomax Distribution with Different Methods of Estimation and Applications,” Complexity, vol. 2020, pp. 1–15, 2020, doi: 10.1155/2020/1860813.

S. Chouia and H. Zeghdoudi, “The X-Lindley distribution: properties and application,” Journal of Statistical Theory and Applications, vol. 20, no. 2, pp. 318–327, 2021.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM Journal on Optimization, vol. 9, no. 1, pp. 112–147, 1998, doi: 10.1137/S1052623496303470.

M. A. Baig et al., “Evaluation and Projection of Temperatures Over Pakistan: Insights from the Downscaled NEX-GDDP-CMIP6 Models,” Earth Systems and Environment, 2025, doi: 10.1007/s41748-024-00554-2.

Z. Jiang, W. Li, J. Xu, and L. Li, “Extreme precipitation indices over China in CMIP5 models. Part I: Model evaluation,” Journal of Climate, vol. 28, no. 21, pp. 8603–8619, 2015, doi: 10.1175/JCLI-D-15-0099.1.

L. D. Ribeiro-Reis, “The kagebushin-beta distribution: an alternative for gamma, Weibull and exponentiated exponential distributions,” Journal of the Egyptian Mathematical Society, vol. 30, no. 1, 2022, doi: 10.1186/s42787-022-00158-7.

M. Farooq, M. Shafique, and M. S. Khattak, “Flood frequency analysis of river swat using Log Pearson type 3, Generalized Extreme Value, Normal, and Gumbel Max distribution methods,” Arabian Journal of Geosciences, vol. 11, no. 9, 2018, doi: 10.1007/s12517-018-3553-z.

W. Szulczewski and W. Jakubowski, “The Application of Mixture Distribution for the Estimation of Extreme Floods in Controlled Catchment Basins,” Water Resources Management, vol. 32, no. 10, pp. 3519–3534, 2018, doi: 10.1007/s11269-018-2005-6.




DOI: http://dx.doi.org/10.30829/zero.v9i3.25692

Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Publisher :
Department of Mathematics
Faculty of Science and Technology
Universitas Islam Negeri Sumatera Utara Medan
📱 WhatsApp:085270009767 (Admin Official)
SINTA 2 Google Scholar CrossRef Garuda DOAJ