ISSN 1794-6190 e-ISSN 2339-3459 https://doi.org/10.15446/esrj.v29n3.119576 EARTH SCIENCES RESEARCH JOURNAL Earth Sci. Res. J. Vol. 29, No. 3 (September, 2025): 363 - 378 M ET EO R O LO G Y Fitting of Statistical Distribution in Assessing Rainfall Patterns Across Various Zones of Assam, India Tanusree Deb Roy1, Dipanjali Ray1*, Subhankar Dutta2, and Sebul Islam Laskar3 1. Department of Statistics, Assam University, Silchar, Assam, India. 2. Department of Mathematics, Bioinformatics and Computer Applications, Maulana Azad National Institute of Technology Bhopal, Madhya Pradesh, India 3. India Meteorological Department (IMD), New Delhi, India * Corresponding author: dipanjali.ray@aus.ac.in Record Manuscript received: 30/03/2025 Accepted for publication: 15/09/2025 How to cite this item: Deb Roy, T., Ray, D., Dutta, S., & Laskar, S. I. (2025). Fitting of Statistical Distribution in Assessing Rainfall Patterns Across Various Zones of Assam. Earth Sciences Research Journal, 29(3), 363-378. https:// doi.org/10.15446/esrj.v29n3.119576 ABSTRACT: The maximum and minimum projected rainfall for a specific time of the year, taking into account a particular level of probabilities, makes up the probable rainfall, which is a great meteorological parameter of information. The purpose of this study is to assess the performance of probability distributions using various goodness-of-fit tests in order to develop a standard for selecting them in various zones of Assam based on monthly rainfall data from 1985-2022. Ten different distributions such as Gumbel, Weibull, Gamma, Logistic, Exponential, log-Normal, Pearson-0, Pear- son-I, Pearson-III and Pearson-V distributions have been considered in the study. The maximum likelihood estimation approach was used to estimate the parameters associated with these distributions. The best fitted probability distri- bution is determined using a variety of goodness of fit techniques, including the Bayesian information criteria (BIC), Akaike information criterion (AIC), and Kolmogorov-Smirnov (K-S) test. Although not one distribution fits the rain- fall data perfectly for every month, the Pearson-I distribution typically fits the data better than the other distributions most of the time, according to the goodness of fit tools. The data has been collected from the National Data Centre (NDC), India Meteorological Department (IMD), Pune. Keywords: AIC; BIC; K-S test; Maximum Likelihood Estimation; Precipitation. Ajuste de la distribución estadística para evaluar los patrones de precipitaciones en diversas zonas de Assam, India RESUMEN La precipitación máxima y mínima proyectada para una época específica del año, con un nivel particular de proba- bilidades, constituye la precipitación probable, que es un gran parámetro meteorológico de información. El propósito de este estudio es evaluar el rendimiento de las distribuciones de probabilidad utilizando varias pruebas de bondad de ajuste con el fin de desarrollar un estándar para luego aplicarlo en varias zonas de Assam con base en datos mensuales de precipitación de 1985 a 2022. En el estudio se han considerado diez distribuciones diferentes como Gumbel, Weibull, Gamma, Logística, Exponencial, log-Normal, Pearson-0, Pearson-I, Pearson-III y Pearson-V. El enfoque de estimación de máxima verosimilitud se utilizó para estimar los parámetros asociados con estas distribuciones. La distribución de probabilidad mejor ajustada se determina utilizando una variedad de técnicas de bondad de ajuste, incluyendo los cri- terios de información bayesianos (BIC), el criterio de información de Akaike (AIC) y la prueba de Kolmogorov-Smir- nov (K-S). Aunque ninguna distribución se ajusta perfectamente a los datos de precipitación para todos los meses, la distribución Pearson-I suele ajustarse mejor a los datos que las demás distribuciones la mayor parte del tiempo, según las herramientas de bondad de ajuste. Los datos se recopilaron del Centro Nacional de Datos (NDC) del Departamento Meteorológico de la India (IMD) en Pune. Palabras clave: AIC; BIC; prueba K-S; precipitación máxima proyectada; precipitation. https://doi.org/10.15446/esrj.v29n3.119576 mailto:dipanjali.ray@aus.ac.in https://doi.org/10.15446/esrj.v29n3.119576 https://doi.org/10.15446/esrj.v29n3.119576 364 Tanusree Deb Roy, Dipanjali Ray, Subhankar Dutta, and Sebul Islam Laskar 1 Introduction: The identification of a probability distribution that precisely matches rainfall data has garnered interest in several disciplines, including hydrology and meteorology. In precipitation research, one of the most prominent topics is determining the most suitable distribution for rainfall data. Various forms of precipitation include drizzle, snow, graupel, sleet, and freezing drizzle, with rainfall being the largest type. By employing several probability distributions, it is feasible to predict the amount of precipitation rather precisely. When managing water resources and identifying the dry and wet seasons, this is crucial information for agricultural planners. Hydrological data sets can be fitted with a variety of probability distributions (Ozonur et al., 2021, Mandal and Choudhury, 2014). Numerous researchers have employed the probability distribution function to examine rainfall in various dimensions. Thom (1958) has analysed the meteorological properties of the Gamma distribution, Bobee and Robitaille (1977) demonstrated that Pearson III outperforms Log Pearson III for flood data across global stations, including 11 stations in Canada, while Swift and Schreuder (1981) evaluated several distributions such as Log-normal, Gamma, Weibull, SB (Special bound), and Beta on daily precipitation data, concluding that the SB distribution best fits high-precipitation areas like the Southern Appalachian Mountains. Sarker et al. (1982) used the Gamma distribution for precipitation in India, Phien and Jivajirajah (1984) applied Log-Pearson III for flood analysis in Thailand, Kottegoda (1987) demonstrated Johnson SB (Special bound) for rainfall in various station of United Kingdom and South Africa, and Koutrouvelis and Canavos (1999) studied Pearson III for 100- year simulated floods data. Griffis and Stedinger (2007) studied Log-Pearson III for U.S. floods, Feng et al. (2007) modeled extreme precipitation in China using the Generalized Extreme Value distribution, and Olofintoye et al. (2009) compared several distributions, including Gumbel and Pearson, for rainfall in 20 Nigerian cities. Deka et al. (2009) analysed the fitting of five three-parameter extreme value distributions i.e. Generalized Extreme Value, Generalized Logistic, Generalized Pareto, Log-Normal, and Pearson III using L-moments and LQ- moments for stations in North-East India maximum daily rainfall data for the period 1966 to 2007. Millington et al. (2011) compared Gumbel, Generalized Extreme Value, and Log-Pearson III distributions for the Upper Thames River, Mandal and Choudhury (2015) identified optimal rainfall distributions for Sagar Island, and Amin et al. (2016) assessed Normal, Log-Normal, Log- Pearson III, and Gumbel Max distributions for northern Pakistan. Kumar et al. (2017) tried to fit Exponential, Gamma, Weibull, Log-Pearson III, and Generalized Extreme Value distributions to rainfall data in Uttarakhand, Douka and Karacostas (2017) evaluated Generalized Pareto, Johnson SB, Log-Gamma, and Log-Normal distributions for precipitation of Thessaloniki. Ghosh et al. (2016) assessed monthly rainfall in Bangladesh using Normal, Lognormal, Gamma, Weibull, Inverse Gaussian, and Generalized Extreme Value distributions. Mamoon and Rahman (2017) examined rainfall frequency in Qatar with fourteen distributions. Bhavyashree and Bhattacharyya (2018) analyzed monsoon rainfall across Karnataka, while Kurniawan (2019) focused on Jakarta’s rainfall patterns. Shaharudin et al. (2020) investigated various distributions for rainfall simulation, and Moccia et al. (2021) evaluated daily rainfall extremes in Lazio and Sicily, Italy, using Weibull, Gumbel, Frechet, Pareto, Gamma, and Lognormal distributions. Greece and Ozonur et al. (2021) tested eight distributions, including Weibull, Rayleigh, Gamma, Log-Normal, and Gumbel, for modelling rainfall data in central West Brazil. Mohamed and Adam (2022) identified the best-fit distribution for maximum rainfall in Somalia, while Chandran et al. (2023) assessed annual, monthly, and seasonal rainfall in Tamil Nadu. Karami et al. (2023) examined distribution models for annual and 24- hour maximum rainfall. Abreu et al. (2023) explored criteria for selecting suitable distributions in case of rainfall data, and Haseeb et al. (2025) compared nine distributions to determine the most appropriate fit for 42 years of rainfall data in Pakistan. Rainfall varies significantly across different locations and times, making statistical distributions helpful in understanding its unpredictability. Examining rainfall data with various statistical models allows researchers to assess variability and identify normal and extreme patterns. Heavy rainfall can lead to flooding, while insufficient rain may lead to droughts. Statistical modelling helps estimate the probability of these extreme events, which is vital for designing flood barriers, building dams, and preparing for drought scenarios. Assam remains highly vulnerable to floods and river basin–related disasters, which exert profound impacts on agriculture and livelihoods. Understanding rainfall patterns is therefore critical, as it can generate valuable insights for flood preparedness, agricultural planning, and regional resilience. However, a comprehensive distributional analysis of rainfall behaviour across Assam for the period 1985–2022 has not yet been undertaken. This study addresses this gap by examining rainfall patterns in the region, thereby contributing to improved climate risk assessment and resource management. This study aims to enhance the understanding of rainfall variability in specific areas of Assam, India. The available climatological data will be analyzed to identify the most suitable probability distribution for rainfall in this region. In this study, ten different distributions have been taken into account to fit the rainfall data. To estimate the parameters of the chosen distributions, the maximum likelihood method has been considered. Before statistical analysis, the Kolmogorov-Smirnov (K-S) test has been considered to assess the adequacy of these distributions to the data based on the K-S distance and the associated p-value. 2 Materials and methods 2.1 Characterization and location of the study area The North East region of India, which includes Assam is expected to be extremely vulnerable to the effects of climate change because of its delicate geo-ecological configuration, strategic location near an international border, the presence of the Eastern Himalayan ranges, transboundary river systems, the inhabitation of the ecosystem by people from various ethnic groups, and inherent socioeconomic disparities (ASTEC, 2011). Depending on its geographical location, the whole Assam is divided into various zones. Namely, Lower Brahmaputra valley zone, Upper Brahmaputra valley zone, North Bank plain zone, Hills zone and Barak valley zone. Five stations have been selected from the five zones except hills zone on the basis of available data provided by meteorological department for the period of 1985- 2022. The name of the stations are given below. Table 1 represents the names of rainfall monitoring stations across various zones of Assam, where the best probability distribution function will be determined from a selection of ten probability distributions. The geological location of the above stations are represented below Table 1. Name of stations in respective zones Name of Zones Name of stations Lower Brahmaputra Valley Zone Guwahati, Dhubri North Bank Plain Zone Tezpur Upper Brahmaputra Valley Zone Dibrugarh Barak Valley Zone Silchar 365Fitting of Statistical Distribution in Assessing Rainfall Patterns Across Various Zones of Assam, India Figure 1. Study area in Assam Figure 1 aims to represent the inferred geological direction of all selected stations spanning diverse zones of Assam created using QGIS 3.32.0 software. Assam’s climate is mostly humid subtropical, with warm, muggy summers, intense monsoons, and moderate winters. The range of winter temperatures is 100 to 220 degrees Celsius. The summertime temperature ranges from 300 to 360 degrees Celsius. The months of July and August receive the most rainfall in Assam (29 percent of the total rainfall during the South West Monsoon), while September receives 24 percent of the total rainfall. The remaining 66% of the yearly rainfall falls during the Southwest Monsoon (June – September). The state Assam lies between 89046/ - 96001/E longitude and 24003/- 27058/ N latitude and covers an area of 78,438 km² (Guhathakurta et al., 2020)The longitude and latitude of remain stations are given below Table 2 represents all the longitudes and latitudes of selected stations for this study. Table 2. Longitude and latitude of selected stations from various zones of Assam Name of stations Longitude Latitude Guwahati 91° 44′ 45″ E 26° 10′ 20″ N Dhubri ~89° 58′ 44″ E ~26° 1′ 20″ N Tezpur 92° 47′ 59″ E 26° 37′ 59″ N Dibrugarh ~94° 54′ 43″ E ~27° 28′ 22″ N Silchar ~92° 47′ 52″ E ~24° 49′ 38″ N 2.2 Data The data is collected from National Data Centre (NDC), India Meteorological Department (IMD), Pune (http://dsp.imdpune.gov.in/) for the period of 1985-2022. For the analysis, data is arranged monthly and also season wise, namely Winter (January, February), Pre-Monsoon (March, April, May), Monsoon (June, July, August, September) and Post-Monsoon (October, November and December). 2.3 Probability distribution In this experiment, ten different probability distributions are considered to study the monthly rainfall data in five stations of various agro-climatic zones of Assam. Gumbel distribution (two-parameter) (Gumbel, 1941), Weibull distribution (two-parameter) (Weibull, 1939), Gamma distribution (two- parameter) (Bobee & Ashkar, 1991), Logistic distribution (two-parameter) (Jhonson et al., 1995), Exponential distribution (one-parameter) (Ferguson, 1964), log-Normal distribution (two-parameter) (Gaddum,1945), Pearson-0 distribution (two-parameter), Pearson-I distribution (four-parameter), Pearson- III distribution (three-parameter) and Pearson-V distribution (three-parameter) (Pearson, 1895) have been considered in the study. Suppose X is a random variable with shape parameter α,a ,b, scale parameter β and location parameter μ then the probability density function (pdf) and their corresponding cumulative distribution function (cdf) are depicted in Table 3. http://dsp.imdpune.gov.in/ 366 Tanusree Deb Roy, Dipanjali Ray, Subhankar Dutta, and Sebul Islam Laskar Table 3. Name of distributions with its pdf, cdf, domain and parameters Distributions pdf cdf Domain Parameter Exponential αe-αx 1 - αe-αx x ∈(0,∞) α > 0 Gumbel 1 x ∈ ℝ μ > 0, β > 0 Weibull 1 x∈(0,∞) α > 0, β > 0 Gamma 1 , x ∈ (0,∞) α > 0, β > 0 Logistic 1 + 1 1 + x∈(-∞,∞) μ > 0, β > 0 Log-Normal 1 2 ( ) 1 2 + 1 2 2 Where erf is the error function x∈(0,∞) μ∈ (-∞,∞), σ > 0 Pearson 0 1 2 ( ) 1 2 ( ) x∈(-∞,∞) μ > 0, σ > 0 Pearson I 1 | | ( + ) 1 | | ( + ) 1 x ∈ [a,b] Where a &b are any real number a > 0, b > 0 β≠0, 0 < < 1 Pearson III ( ) 1 ( ) 𝑥 ∈ [𝜇,∞] 𝑎 > 0, 𝛽 > 0 > 0 Pearson V | | | | | | | | 𝑥 ∈ (0,∞) 𝑎 > 0, 𝛼,𝛽 >0 >0 2.4 Maximum likelihood method The parameter estimates of the distributions may be estimated using a variety of techniques. The maximum likelihood approach is the most used estimating technique. This approach is used in this work to estimate the parameters of any probability distribution that is taken into consideration. Due to its good asymptotic features, this approach is a widely used and recommended statistical estimating technique. It is widely acknowledged that it generates parameter estimates that are both statistically consistent and effective. If we consider a sample {X1,X2,…,Xn } following distributions with PDF f(x;ϴ), where ϴ represents the set of parameters, then the log-likelihood function can be expressed as ( , , … , ; ) = log ( ; ) . By maximizing the log L, , the maximum likelihood estimates (MLEs) can be obtained. To obtain the MLEs, we have to solve = 0 , with respect to set of parameters. Sometimes these equations cannot be solved explicitly, then Newton-Raphson method is used (Gupta & Kapoor, 1997; Feng et al., 2007; Ozonur et al., 2021). 2.5 Goodness of fit tests and model selection criteria To assess the appropriateness of the chosen probability distributions to the rainfall data, Kolmogorov-Smirnov (K-S) test has been considered initially. Additionally, Akaike’s information criterion (AIC) and Bayesian information criterion (BIC) have been considered as the model selection criteria. Using K-S test, one can determine whether the data follows the considered distribution or not. In K-S test, the null hypothesis has been considered as the data follows the selected distribution whereas the alternative hypothesis yields that the data does not follow the selected distribution. K-S test statistic defines the maximum difference between sample and theoretical CDF, and it can be expressed as = max ( ) , ( ) 1 , 367Fitting of Statistical Distribution in Assessing Rainfall Patterns Across Various Zones of Assam, India Where x((k)) is the estimated value of the CDF, and {x(1) ,…,x(n)} are ascending ordered observations. If the test statistics Dn > Dn (α) then the null hypothesis will be rejected at α significance level, where Dn(α) represents the critical value of the K-S statistic (An & Cheng, 1996; Sharma and Singh, 2010; Mandal and Choudhury, 2014; Ozonur et al., 2021; Mamoon & Rahman, 2017; Moccia et al. 2021). Further, the p-values of the goodness of tests are often used to assess the appropriateness of the above-mentioned distributions. By using the p-value technique, the null hypothesis is rejected if the p-value is less than the selected significance threshold. The null hypothesis is not rejected if the p-value is bigger than the significance threshold, and there is insufficient evidence to draw the conclusion that the data do not follow the given distribution. If it is determined that both of the distributions under comparison may be appropriate, then the distribution with the greater p-value fits the data more closely. The effectiveness of the investigated pdfs for modelling the rainfall data is determined using the model selection criteria AIC and BIC after the appropriateness of the pdfs is determined based on the findings of the goodness of fit tests. The AIC and BIC can be obtained by using the following formulas: AIC = -2 log L + 2m, BIC = -2 log L + m log n , where log L yields the value of the estimated log-likelihood function at the MLEs of the selected model, m is the number of parameters of the selected distribution, and n is the number of observed data (Akaike, 1969; Schwarz, 1978; Ozonur et al., 2021; Mohamad & Adam, 2021). These are analysed in software R version 4.1.1. 3. Result and discussion In this section, the best fit distribution is elaborately discussed on total rainfall (mm) monthly and season wise for Guwahati, Silchar, Tezpur, Dibrugarh, Dhubri station and are given below The above Table 4 describes the frame of the data in both monthly and season wise by introducing length of the data, mean, maximum (Max), minimum (Min), median, standard deviation (SD), coefficient of variation (CV), coefficient of skewness (CS) and coefficient of kurtosis (CK). Out of the twelve months, on April month variation is highest as SD is 122.444 and in season wise on Pre-Monsoon season shows highest variation (SD=124.134). Similarly, the descriptive statistics have been calculated for the remaining station from which it has been observed that April (194.319) month and Pre- Monsoon (201.759) show highest SD in Silchar station. The month of June is showing highest SD for the remaining stations i.e. Tezpur (144.872), Dibrugarh (165.55) and Dhubri (281.681). Season wise, Monsoon shows highest SD for the remaining station Tezpur (118.590), Dibrugarh (149.852) and Dhubri (233.280). Table 4. Descriptive statistics of rainfall in monthly and season wise data of Guwahati station for the period of 1985-2022 Month Length Mean Min Max Median SD CV CS CK January 38 11.387 0 78.100 7.900 15.104 132.648 2.416 7.599 February 38 19.840 0 96.100 14.700 19.812 99.850 1.707 3.744 March 38 52.180 3.800 152.100 42.450 40.544 77.703 0.891 -0.168 April 38 179.200 24.000 520.900 160.700 102.342 57.121 0.889 1.431 May 38 246.200 88.500 569.700 222.000 122.444 49.729 0.777 -0.120 June 38 301.600 104.100 573.000 307.200 117.190 38.853 0.262 -0.825 July 38 301.900 131.600 638.800 285.900 118.365 39.207 0.880 0.397 August 38 239.100 67.100 569.200 221.100 111.544 46.660 0.550 0.080 September 38 190.300 28.200 372.000 183.100 86.267 45.342 0.353 -0.786 October 38 115.540 7.200 354.400 117.750 81.680 70.691 0.868 0.618 November 38 11.900 0 88.100 4.450 18.381 154.463 2.367 6.005 December 38 4.989 0 29.000 0.600 8.014 160.626 1.554 1.155 Winter 76 15.610 0 96.100 11.000 18.006 115.332 2.023 5.272 Pre-Mon 114 159.189 3.800 569.700 135.100 124.134 77.978 1.008 0.755 Monsoon 152 258.200 28.200 638.800 242.100 117.810 45.626 0.623 0.177 Post-Mon 114 44.140 0 354.400 8.650 69.969 158.501 2.054 4.315 368 Tanusree Deb Roy, Dipanjali Ray, Subhankar Dutta, and Sebul Islam Laskar Table 5. Estimates of probability distribution fitting of total monthly rainfall in Guwahati station for January, February and March month Estimates Months Distribution Shape1 Shape2 Location Scale January Gamma 0.2525 - - 0.0221 Logistics - - 9.1336 6.9780 Exponential 0.0878 - - - Gumbel - - 5.7015 8.4914 Weibull 0.3485 - - 5.0727 Log-Normal - - 0.3758 4.7371 Pearson 0 - - 11.3868 14.9043 Pearson I 0.4280 8.4427 0.0001 223.9029 Pearson III - 0.6068 0.0001 17.3831 Pearson V - 2.09E-01 1.00E-04 1.95E-14 February Gamma 0.4177 - - 0.0210 Logistic - - 17.2762 9.9634 Exponential 0.0504 - - - Gumbel - - 11.7239 12.7800 Weibull 0.5721 - - 15.6152 Log-Normal - - 1.4234 3.7790 Pearson 0 - - 19.8421 19.5499 Pearson I 0.7961 13.3457 0.0001 320.231 Pearson III - 0.9681 0.0001 16.1597 Pearson V - 3.1893 -10.2752 68.1229 March Gamma 1.5351 - - 0.0294 Logistic - - 47.3156 22.5857 Exponential 0.0191 - - - Gumbel - - 34.2173 28.9744 Weibull 1.3049 - - 56.5877 Log-Normal - - 3.5949 0.9391 Pearson 0 - - 52.1789 40.0078 Pearson I 0.7492 1.918 3.8 173.9859 Pearson III - 0.9922 3.8 47.0663 Pearson V - 36.4221 -166.352 7729.309 Table 5 represents all the estimates of parameters i.e. shape parameter α,a ,b, , scale parameter β and location parameter μ for January, February and March month considering rainfall data for the period of 1985-2022 in Guwahati station. Table 6 represents probability distribution fitting of monthly total rainfall in Guwahati station monthly for January, February and March month. It elaborates parameters for each distribution and comparison criteria’s i.e. -2logL (log likelihood), AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), K-S value and p-value. Fitting of best probability distribution on a data set may be characterised as finding the distribution with the lowest values of -2logL, AIC, BIC, and K-S values. Here p-value plays very significant role, since by having greater value than 0.05 represents the data follows the required distribution. The findings of the K-S test indicate that the rainfall data for January at the Guwahati station does not follow the Gamma, Exponential, Log-Normal, or Pearson 0 and V distributions, as their p-values are less than 0.05. In contrast, the data does conform to the Logistic, Weibull, Pearson Type I, and Pearson Type III distributions since their p-values are greater than 0.05. Among the followed distribution by January month, Pearson I distribution has least -2logL (14.5150), AIC (22.1151), BIC (29.0654), K-S value (0.2105) and p-value is 0.0589, which is greater than 0.05 which indicates that the data points satisfy the null hypothesis i.e. the data points of January month follow Pearson I distribution. Likewise, interpretation can be elaborated for remaining months also. 369Fitting of Statistical Distribution in Assessing Rainfall Patterns Across Various Zones of Assam, India Table 6. Criteria for comparison of probability distribution fitting of total monthly rainfall in Guwahati station for January, February and March Criteria for Comparison Months Distribution -2logL AIC BIC KS-value p-value January Gamma 168.0214 172.0214 175.2966 0.2213 0.0482 Logistic 302.6070 306.607 309.8821 0.2126 0.0643 Exponential 260.8668 262.8668 264.5044 0.2479 0.0187 Gumbel 289.4652 293.4652 296.7404 0.1681 0.2331 Weibull 180.5210 184.521 187.7962 0.217 0.0557 Log-Normal 197.4897 201.4897 204.7649 0.2482 0.0185 Pearson 0 313.1651 317.1651 320.4403 0.2224 0.0392 Pearson I 14.5150 22.1151 29.0654 0.2105 0.0589 Pearson III 70.9679 76.9679 81.8807 0.2105 0.0589 Pearson V 164.1624 170.1624 175.0751 0.7874 2.22E-16 February Gamma 274.4866 278.4867 281.7618 0.2197 0.051 Logistic 328.2530 332.2529 335.5281 0.1500 0.3589 Exponential 303.0732 305.0733 306.7109 0.1052 0.7937 Gumbel 317.9188 321.9187 325.1939 0.1173 0.6723 Weibull 284.239 288.2391 291.5143 0.2051 0.0815 Log-Normal 317.0596 321.0596 324.3348 0.3241 0.0006 Pearson 0 333.7851 337.7851 341.0603 0.1557 0.2894 Pearson I 255.954 263.954 270.5044 0.1200 0.6014 Pearson III 297.6733 303.6733 308.586 0.1567 0.2775 Pearson V 312.9186 318.9186 323.8314 0.1008 0.7974 March Gamma 372.8106 376.8107 380.0859 0.0763 0.9676 Logistic 388.5022 392.5022 395.7774 0.1271 0.5295 Exponential 376.5556 378.5556 380.1932 0.1196 0.6061 Gumbel 378.9424 382.9424 386.2176 0.0952 0.8484 Weibull 372.7302 376.7303 380.0055 0.0723 0.9803 Log-Normal 376.2848 380.2848 383.5600 0.0997 0.8085 Pearson 0 388.2091 392.2091 395.4843 0.1630 0.2372 Pearson I 354.6303 362.6303 369.1807 0.0914 0.8794 Pearson III 370.4733 376.4733 381.3861 0.1124 0.6805 Pearson V 381.5386 387.5386 392.4513 0.11345 0.6704 The criteria for fitting probability distribution functions, along with their best parameter estimates, have been thoroughly discussed above for each month, supported by statistical values to demonstrate their significance. The graphical representation of these distributions for any given month or season can be effectively illustrated using a histogram of the raw data from the corresponding period, overlaid with line plots for each fitted distribution. In line with this approach, the rainfall data for January of Guwahati station can be depicted graphically below, and this procedure can similarly be applied to the remaining months and seasons for the station to provide a comprehensive visual analysis. Figure 2 represents the histogram with multiple line diagrams of ten selected probability distribution functions on raw rainfall data of January month in Guwahati station just to check the suitability of these selected distributions on rainfall data. It is distinctly observed that these distributions effectively characterize the rainfall data for January month of Guwahati station, and this approach can be systematically extended to encompass the remaining months and the rainfall records of other stations also. Figure 3 represents a Q-Q plot, which shows how the quantiles of two distributions relate, indicating if the data follow a specific theoretical distribution or share the same distribution. Gumbel (orange) fits reasonably well in the middle but overestimates at higher quantiles, with points above the line. Logistic (blue) outperforms exponential and gamma, especially in the mid to upper range, though some deviations remain at the extremes. Weibull (brown) stays close to the line in lower to mid quantiles but diverges upward at higher values. P1/Pearson I (orange) aligns better than Log-Normal (LGN) and Pearson 0 (P0), but still deviates in mid-probabilities. P3/Pearson III (blue) remains closer to the line across most of the range, especially mid to upper probabilities, indicating a better fit. These visual representations can also be summarised by test statistics. This visual interpretation is nearly the same as the interpretation of Table 6, which shows various criteria for the comparisons of the selected distribution. 370 Tanusree Deb Roy, Dipanjali Ray, Subhankar Dutta, and Sebul Islam Laskar Figure 4 represents a P-P plot which compares the empirical cumulative distribution function of January data to a theoretical distribution’s cumulative function. Its interpretation is similar to that of a Q-Q plot. The logistic distribution (blue) follows the reference line more closely, especially in the central part, but shows divergence in the tails. The Weibull distribution (brown) stays near the diagonal over a wide range, although it doesn’t fit perfectly in the lower tail. P3/Pearson III (blue) remains closest to the diagonal throughout most of the range and provides the best fit overall among the second set of options. Figure 2. Histogram with multiple line diagrams of 10 selected distributions of January month of Guwahati station for the period of 1985-2022 on Rainfall Figure 3. Q-Q Plot of January month at Guwahati Station Figure 4. P-P Plot of January month at Guwahati Station 371Fitting of Statistical Distribution in Assessing Rainfall Patterns Across Various Zones of Assam, India Table 7. Criteria for comparison of best probability distribution fitting of total monthly rainfall in Guwahati station for April, May, June, July, August, September, November and December Station: Guwahati Months Distribution -2logL AIC BIC KS-value p-value April Pearson I 442.3452 450.3452 456.8956 0.2426 0.1870 May Pearson I 454.6502 462.6502 469.2005 0.1205 0.5967 June Pearson I 463.8386 471.8386 478.389 0.0786 0.9580 July Log-Normal 462.3650 466.3650 469.6402 0.0596 0.9993 August Weibull 461.8864 465.8865 469.1616 0.0726 0.9881 September Weibull 443.2952 447.2953 450.5704 0.0639 0.9977 October Weibull 431.2786 435.2786 438.5538 0.1565 0.3098 November Gamma 178.507 182.507 185.7822 0.1778 0.1808 December - - - - - - Table 8. Criteria for comparison of probability distribution fitting of total monthly rainfall in Guwahati station for Winter, Pre-Monsoon, Monsoon and Post-Monsoon season Season wise (Guwahati Station) Months Distribution -2logL AIC BIC KS-value p-value Winter Gamma 448.0652 452.0652 456.7266 0.2239 0.0009 Logistic 639.2842 643.2842 647.9457 0.1863 0.0102 Exponential 569.7262 571.7261 574.0569 0.1737 0.0203 Gumbel 615.1228 619.1229 623.7843 0.1373 0.1135 Weibull 472.7544 476.7545 481.416 0.2188 0.0013 Log-Normal 519.7514 523.7513 528.4128 0.2828 1.05E-05 Pearson 0 654.0812 658.0812 662.7426 0.1913 0.0065 Pearson I 267.4619 275.4619 284.7848 0.1578 0.0402 Pearson III 374.2139 380.2139 387.2061 0.1627 0.0315 Pearson V 593.8993 599.8993 606.8915 0.14447 0.0747 Pre-Mon Pearson I 1371.428 1379.428 1390.372 0.0751 0.5159 Monsoon Pearson I 1865.889 1873.889 1885.984 0.0469 0.8755 Post-Mon - - - - - - From Table 6 and 7, the data points of February, March, April, May, June month fit best the Pearson I distribution with least -2logL, AIC, BIC and K-S value with greater 0.05 p-value, but in July month, both Pearson I and Log-normal distribution fits well, but it is more favored to choose Log- normal distribution since K-S value is lowest in case of Log-normal distribution than Pearson I distribution. Similarly, August, September and October month follow Weibull distribution. November month follows Gamma distribution and December month doesn’t follow any distribution i.e. the p-value for each distribution satisfies the alternative hypothesis of K-S test. It means none of 10 distributions follow the required distribution i.e. all are having p-values less than 0.05. The criteria for determining the best-fitting probability distribution season wise are presented in Table 8. Table 8 also briefly explains the quantitative values of criteria’s of fitting best distribution in season wise i.e. Winter (January, February), Pre- Monsoon (March, April, May), Monsoon (June, July, August, September) and Post-Monsoon season (October, November, December) respectively. It is observed that the rainfall data of January and February month i.e. Winter season follows Gumbel and Pearson V distribution with greater than 0.05 p-value, but depending on K-S distance or value, Gumbel fits better with least k-S value. Similarly, Pre-Monsoon follows Pearson I, Monsoon follows Pearson I distribution and Post-Monsoon does not satisfy the null hypothesis of K-S test. Table 9, 10, 11 and 12 represents selected criteria’s of fitting of distribution with least quantitative value for each month and seasons for Silchar, Tezpur, Dibrugarh and Dhubri respectively. From Table 5 to 12, it is observed that among 10 distributions, 34.48%, 22.48% and 12.06% of rainfall data from 1985-2022, follows Pearson type 1, Weibull and Gumbel distribution respectively considering net stations in monthly wise. Similarly, 50%, 18.75% and 12.5% follows Pearson type 1, Weibull and Gumbel distribution in season wise also. This analysis provides a refined understanding of probability distribution functions for monthly and seasonal rainfall, essential for shaping advanced agricultural strategies. By employing these models, farmers can accurately predict rainfall patterns, allowing precise adjustments in crop selection, planting schedules, and resource management. This precision enhances operational efficiency while promoting sustainable practices, ensuring resilience and long- term productivity in an increasingly unpredictable climate. 372 Tanusree Deb Roy, Dipanjali Ray, Subhankar Dutta, and Sebul Islam Laskar Table 9. Criteria for comparison of best probability distribution fitting of total monthly rainfall in Silchar station Station: Silchar Criteria for comparison Months Distribution -2logL AIC BIC KS-value p-value January Gumbel 272.3906 276.3906 279.6658 0.2059 0.0795 February Pearson I 279.4234 287.4234 293.9738 0.1315 0.4859 March Weibull 451.964 455.964 459.2392 0.0805 0.9495 April Pearson I 487.2584 495.2584 501.8087 0.1787 0.1558 May Weibull 506.2132 510.2132 513.4883 0.0797 0.9534 June Pearson II 501.6652 509.5792 516.1295 0.0798 0.9530 July Pearson V 486.2718 492.2718 497.1846 0.1070 0.7369 August Logistic 491.0006 495.0006 498.2758 0.0917 0.9061 September Pearson III 485.5180 491.5180 496.4307 0.1172 0.6308 October Gumbel 453.9594 457.9594 461.2346 0.1388 0.4564 November Gumbel 349.366 353.3666 356.6412 0.2133 0.0628 December Weibull 408.8964 412.8965 416.1716 0.1858 0.1447 Season wise Winter Pearson I 280.6986 288.6986 295.3529 0.1282 0.5027 Pre-Mon Pearson I 294.8297 302.8297 309.5852 0.1200 0.5192 Monsoon Logistic 1995.745 1999.745 2005.793 0.0424 0.9467 Post-Mon - - - - - - Table 10. Criteria for comparison of best probability distribution fitting of total monthly rainfall in Tezpur station Station: Tezpur Months Distribution -2logL AIC BIC KS-value p-value January Pearson I 74.2633 82.2633 88.8137 0.1861 0.1263 February Pearson I 293.1259 301.1259 307.6762 0.0677 0.9901 March Pearson I 332.5493 340.5493 347.0996 0.0943 0.8563 April Gamma 411.9816 445.9816 449.2568 0.0917 0.8775 May Weibull 446.7120 450.7119 453.9871 0.0962 0.873 June Pearson 0 484.9906 488.9906 492.2658 0.0835 0.9334 July Gamma 454.6172 458.6172 461.8924 0.0989 0.8511 August Pearson I 452.3756 460.3756 466.9259 0.1027 0.7793 September Pearson I 444.3329 452.3329 458.1402 0.1215 0.6280 October Pearson I 396.8594 404.8594 411.4098 0.0813 0.9453 November Pearson I 94.5341 102.5341 108.8682 0.218 0.0555 December Pearson I 46.8992 54.8992 61.3428 0.1351 0.4684 Season wise Winter Pearson I 399.0063 407.0063 416.3293 0.1032 0.3674 Pre-Mon Pearson I 1352.836 1360.836 1371.78 0.0678 0.6446 Monsoon Gumbel 1879.527 1833.527 1889.575 0.0531 0.7835 Post-Mon Pearson I 588.7287 596.7287 607.5668 0.1084 0.1387 373Fitting of Statistical Distribution in Assessing Rainfall Patterns Across Various Zones of Assam, India Table 11. Criteria for comparison of best probability distribution fitting of total monthly rainfall in Dibrugarh station Station: Dibrugarh Months Distribution -2logL AIC BIC KS-value p-value January Weibull 314.4238 318.4238 321.6989 0.1380 0.4637 February Weibull 385.9224 389.9224 393.1976 0.0857 0.9427 March Pearson I 381.6493 389.6493 396.1997 0.1235 0.5656 April Weibull 314.4238 318.4238 321.6989 0.138 0.4637 May Pearson III 481.0075 487.0075 491.9202 0.1004 0.8015 June Logistic 493.8898 497.8899 501.1651 0.0856 0.9205 July Pearson I 466.1879 474.1879 480.7382 0.0748 0.9726 August Logistic 487.8244 491.8243 495.0995 0.1525 0.3394 September Pearson I 463.8983 471.8983 478.4487 0.1112 0.6937 October Pearson I 411.8757 419.8757 426.426 0.1434 0.3784 November Gumbel 372.4454 376.4454 379.7205 0.1657 0.2477 December Weibull 190.1 194.1 197.3219 0.1949 0.1200 Season wise Winter Weibull 722.0576 726.0576 730.719 0.0617 0.9341 Pre-Mon Gamma 1412.07 1416.0700 1421.542 0.0532 0.9341 Monsoon Pearson III 1953.289 1957.337 1963.337 0.0373 0.9786 Post-Mon Weibull 992.6616 996.6616 1002.081 0.1123 0.1214 Table 12. Criteria for comparison of best probability distribution fitting of total monthly rainfall in Dhubri station Station: Dhubri Months Distribution -2logL AIC BIC KS-value p-value January - - - - - - February Gamma 112.5378 316.5378 319.813 0.2032 0.0864 March Gumbel 401.4268 405.4268 408.7019 0.1792 0.1740 April Gumbel 460.5102 464.5102 467.7853 0.1120 0.7264 May Logistic 477.3674 481.3674 484.6426 0.1262 0.5802 June Weibull 503.5918 507.5918 510.867 0.0749 0.9225 July Pearson V 486.2718 492.2718 497.1846 0.10702 0.7369 August Gumbel 494.8362 498.8362 502.143 0.0944 0.8872 September Gamma 482.6098 486.6098 489.8849 0.1099 0.7477 October Weibull 454.9794 458.9793 462.9793 0.1485 0.3715 November Pearson I 124.111 132.111 138.6614 0.1323 0.4784 December - - - - - - Season wise Winter - - - - - - Pre-Mon Gumbel 1452.097 1461.569 1461.569 0.0779 0.4926 Monsoon Weibull 1501.223 1505.223 1510.696 0.0536 0.8979 Pearson III 1499.329 1505.329 1513.537 0.0556 0.8519 Post-Mon - - - - - - 374 Tanusree Deb Roy, Dipanjali Ray, Subhankar Dutta, and Sebul Islam Laskar 4. Conclusion In this study ten probability distribution functions were considered to fit the monthly and season wise rainfall data of five zones of Assam. From the analysis, the results reveal the statistical frame of data set for each station. Considering all the months and seasons of five stations, the month of June for Dhubri station shows the highest SD (281.681) and Monsoon (233.280) season of Dhubri gives peak of SD which indicates there is a higher deviation in rainfall data in the month of June and in Monsoon (June, July, August, September) season amongst the five stations. The criteria of best fitting of selected distributions are elaborately explained along with the estimation of parameter. The Weibull distribution fits best in the month of January and Pearson V fits best in Winter season considering ten distributions with least -2logL, AIC, BIC, K-S value with a p-value greater than 0.05 in Guwahati station. Likewise, the best fitting distribution along with its criteria of fitting and parameter estimation are explained monthly and season wise for each station. By taking into consideration all the stations in case of best fitting distribution, Pearson type 1, Weibull and Gumbel distribution covers 34.48 percent, 22.48 percent and 12.06 percent respectively for the twelve months. For season wise, the percentage area of Pearson type 1 is 50 percent, 18.75 percent for Weibull and Gumbel distribution covers 12.5 percent. This paper elaborates the fitting of distribution on rainfall data in various zones of Assam which indicates there is also a possibility to get more precise form of distribution by proposing new generalized form of distribution to fit rainfall data in future. Accurate rainfall predictions are vital for estimating river discharge and reservoir inflows, enabling early flood warnings. Short-term forecasts are especially important for managing stormwater drainage. For example, in areas like Guwahati and Silchar, Assam farmers rely heavily on monsoon rainfall. Seasonal weather forecasts guide their decisions on sowing, transplanting paddy, and selecting short-duration crops. Understanding the probability of heavy rain allows farmers to adopt preventive measures against waterlogging, crop damage, pests, and outbreaks diseases etc. Acknowledgements The authors would like to thank the India Meteorological Department (IMD), Pune for supplying the required study’s data. Authors would like to thank the editor and the referees for their support and assistance to improve the manuscript. Author Contributions All authors contributed to the study conception and design. Material preparation was performed by Tanusree Deb Roy and Subhankar Dutta, data collection was done by Sebul Islam Laskar and analysis was performed by Dipanjali Ray. 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Estimates of probability distribution fitting of total monthly rainfall in Guwahati station for remaining months Months Distribution Shape1 Shape2 Location Scale April Pearson I 0.7542 2.1747 24.0000 522.8001 May Pearson I 0.9553 3.1823 88.5000 642.4579 June Pearson I 1.2658 1.8853 100.8439 496.6607 July Log-Normal - - 5.6387 0.3776 August Weibull 2.3195 - - 270.3478 September Weibull 2.3941 - - 214.9069 October Weibull 1.3993 - - 126.3508 November Gamma 0.2627 - - 0.022 December - - - - - Table B. Estimates of probability distribution fitting of total monthly rainfall in Guwahati station for Winter, Pre-Monsoon, Monsoon and Post-Monsoon season Months Distribution Shape1 Shape2 Location Scale Winter Gamma 0.3085 - - 0.0197 Logistic - - 12.9754 8.8017 Exponential 0.0640 - - - Gumbel - - 8.4552 10.9346 Weibull 0.4250 - - 9.4203 Log-Normal - - 0.5238 4.3783 Pearson 0 - - 15.6144 17.8897 Pearson I 0.5949 10.2068 0.0001 266.7931 Pearson III - 0.7298 0.0001 17.0653 Pearson V - 1.5629 -3.9785 14.8985 Pre-Mon Pearson I 0.9917 4.8964 3.8000 911.8026 Mon Pearson I 3.4559 11.5491 1.8968 1113.133 Post-Mon - - - - - In Table A, the parameters for April to December at the Guwahati station are listed. Similarly, in Table B, each parameter for each season i.e. Winter (January, February), Pre-Monsoon (March, April, May) and Monsoon (June, July, August, September) are mentioned except Post-Monsoon season (October, November and December), since Post-Monsoon season does not follow any one of the 10 selected distributions with the rainfall data in Guwahati Station. For the winter season, parameter estimates are presented for all ten selected distributions, whereas for the other seasons, only the parameters of the best- fitted distribution are reported. 377Fitting of Statistical Distribution in Assessing Rainfall Patterns Across Various Zones of Assam, India Appendix B: Parameter Estimates of Silchar Station, Tezpur Station, Dibrugarh Station and Dhubri Station in monthly and season wise data. Table C: Estimates of best probability distribution fitting of total monthly rainfall in Silchar Station Station: Silchar Estimates Months Distribution Shape1 Shape2 Location Scale January Gumbel - - 3.9666 6.6712 February Pearson I 0.3509 0.9882 0.0001 0.0652 March Weibull 1.3377 - - 162.4325 April Pearson I 0.8518 1.8127 23.2000 802.9113 May Weibull 2.2891 - - 468.592 June Pearson II 1.7622 - 109.7549 803.5272 July Pearson V - 5.6109 492.3355 120.8108 August Logistic - - 406.7817 85.6269 September Gamma - 6.4457 - 0.0173 October Gumbel - - 136.06 82.2392 November Gumbel - - 14.0195 19.143 December Weibull - 0.4074 - 95.1627 Season wise Winter Pearson I 0.3621 1.0518 0.0001 659.5552 Pre-Mon Pearson I 0.3588 1.0380 0.0001 657.548 Monsoon Logistic - - 443.0995 96.6872 Post-Mon - - - - - Table D. Estimates of best probability distribution fitting of total monthly rainfall in Tezpur Station Station: Tezpur Months Distribution Shape1 Shape2 Location Scale January Pearson I 0.2866 1.2953 0.0001 101.5330 February Pearson I 0.6298 2.1496 0.0001 110.4932 March Pearson I 0.6321 0.8496 0.9000 109.7770 April Gamma 3.4355 - - 0.0205 May Weibull 3.1960 - - 283.2683 June Pearson 0 - - 303.8474 142.9531 July Gamma 9.4320 - - 0.0308 August Pearson I 0.9232 1.3029 97.5000 425.9379 September Pearson I 0.9534 2.3509 85.5000 475.2478 October Pearson I 0.8165 1.6495 12.0000 254.0389 November Pearson I 0.5354 3.7419 0.0001 147.7994 December Pearson I 0.3106 3.5235 0.0001 133.2491 Season wise Winter Pearson I 0.3837 1.6437 0.0001 103.9733 Pre-Mon Pearson I 0.9077 2.0101 0.9000 482.7833 Monsoon Gumbel - - 220.4292 103.0446 Post-Mon Pearson I 0.3455 1.7339 0.0001 278.8994 378 Tanusree Deb Roy, Dipanjali Ray, Subhankar Dutta, and Sebul Islam Laskar Table E. Estimates of best probability distribution fitting of total monthly rainfall in Dibrugarh Station Station: Dibrugarh Months Distribution Shape1 Shape2 Location Scale January Weibull 1.2982 - - 26.0708 February Weibull 1.2670 - - 65.7468 March Pearson I 0.6223 0.8717 28.8000 211.9777 April Weibull 1.2982 - - 26.0708 May Pearson III - 3.3481 29.1147 82.5268 June Logistic - - 396.9677 90.1618 July Pearson I 4.572 4.7715 639.4236 245.1121 August Logistic - - 373.2674 82.0850 September Pearson I 0.9131 1.3561 128.1000 500.7443 October Pearson I 0.8149 1.5532 30.6000 289.403 November Gumbel - - 13.5589 22.0851 December Weibull 1.9822 - - 5.7701 Season wise Winter Weibull 1.0804 - - 44.029 Pre-Mon Gamma 2.3473 - - 0.0110 Monsoon Pearson III - - 397.9645 149.3592 Post-Mon Weibull 0.4773 - - 32.7641 Table F. Estimates of best probability distribution fitting of total monthly rainfall in Dhubri Station Station: Dhubri Months Distribution Shape1 Shape2 Location Scale January - - - - - February Gamma 0.2849 - - 0.0048 March Gumbel - - 31.0939 37.3340 April Gumbel - - 141.1324 88.0224 May Logistic - - 324.4702 73.7592 June Weibull 3.1264 - - 579.8261 July Pearson VII - 5.6109 492.3355 120.8100 August Gumbel - - 248.8648 137.3011 September Gamma 6.4457 - - 0.0173 October Weibull 1.8043 - - 204.1932 November Pearson I 0.3539 0.7547 0.0001 94.8928 December - - - - - Season wise Winter - - - - - Pre-Mon Gumbel - - 120.9820 117.9553 Monsoon Weibull 2.649 - - 494.7737 Pearson III - 25.8363 458.5850 34.6120 Post-Mon - - - - - Table C, D, E and F represents the estimates of parameters of best distribution among ten selected distribution considering rainfall data for each month and season wise for Silchar, Tezpur, Dibrugarh and Dhubri station respectively. _Hlk165186881 _Hlk165187188 _Hlk174727923 _GoBack _Hlk174389375