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International Journal of Current Microbiology and Applied Sciences ISSN: 2319-7706 Volume 9 Number 2 (2020) Journal homepage: http://www.ijcmas.com

Original Research Article https://doi.org/10.20546/ijcmas.2020.902.339

Exploring Different Probability Distributions for Rainfall Data of Kodagu - An Assisting Approach for Food Security

R. Shreyas1*, D. Punith1, L. Bhagirathi2, Anantha Krishna3 and G. M. Devagiri4

1UAHS (Shivamogga), College of Forestry,Ponnampet, Karnataka-571216, India 2Department of Basic Sciences, College of Forestry, Ponnampet, Karnataka-571216, India 3Department of Computer Science, College of Forestry, Ponnampet, Karnataka-571216, India 4Department of Natural Resource Management, College of Forestry, Ponnampet, Karnataka-571216, India

*Corresponding author

ABSTRACT

Rainfall intensity, duration and its distribution play a major role in the growth of agriculture and other related sectors and the overall development of a country. The present study is carried out to know the best fitting

K e yw or ds probability distribution for rainfall data in three different taluks of Kodagu

Rainfall, probability District. The time series data of average monthly and annual rainfall over a distributions, fitting, period of 61 years (1958-2018) was collected from KSNDMC, Bangalore. goodness-of-fit Around 26 different probability distributions were used to evaluate the best

Article Info fit for annual and seasonal rainfall data. Kolmogorov-Smirnov, Anderson Darling and Chi-squared tests were used for the goodness of fit test. The Accepted: 20 January 2020 best fitting distribution was identified by maximum score which is a sum of Available Online: ranks given by three selected goodness of fit test for the distributions which 10 February 2020 is again based on fitting distance. Among various distributions attempted-

Log Logistic (3P), Dagum, Gamma (3P), Inverse Gaussian, Generalized Gamma, Pearson Type 5 (3P) and Pearson 6 were found to be the best fit for annual and seasonal rainfall for different taluks of Kodagu district.

Introduction amount of rainfall during the monsoon season is very important for economic activity. Indian agriculture sector accounts for around 14 percent of the country’s economy but Rainfall intensity, duration and its distribution accounts for 42 percent of total employment play a major role in the growth of agriculture in the country. About 55 percent of India’s and other related sectors and the overall arable land depends on precipitation, the development of a country. Rainfall intensity,

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Int.J.Curr.Microbiol.App.Sci (2020) 9(2): 2972-2980 patterns and its distribution are altered by medicine, climatology, economics and natural climatic variability i.e., decadal agricultural science. Probability distributions changes in circulation (Deepthi K.A, 2015) as of rainfall have been studied by many well as human induced changes i.e., land use researchers. and cover, emission of greenhouse gases, etc. The variability in rainfall affects the The main objective of this study is to identify agricultural production, water supply, a suitable probability distribution for annual transportation, the entire economy of a region, and seasonal rainfall in the different taluks of and the existence of its people. In regions Kodagu. where the year-to-year variability is high, people often suffer great calamities due to Materials and Methods floods or droughts. The damage due to extremes of rainfall cannot be avoided Kodagu district with an area of 4102 km sq. is completely, a forewarning could certainly be one of the smallest districts in the state of useful and it’s possible from analysis of Karnataka, located between 11056’00’’ and rainfall data. 12050’00’’ North latitude and between 75022’00” and 76011’00” East longitude. The In India, the monsoon or rainy season is average annual rainfall is around 2682 mm dominated by the humid South West (Anonymous, 2018). The District is composed Monsoon that sweeps across the country in of three taluks namely Madikeri, Somwarpet early June, first hitting the State of Kerala. and Virajpet. The southwest monsoon is generally expected to begin in early June and end by September Data and the total rainfall of these four months is considered as monsoon rainfall. In Indian Rainfall of three taluks of Kodagu district was agriculture, the contribution of south-west selected with annual and seasonal series from monsoon is immense as more than 70% of 1958 to 2018. The required rainfall data for India’s annual rainfall is from the south west the study was collected from Karnataka State monsoon and supports nearly 75% of the Natural Disaster Monitoring Centre kharif crop which is critical to India’s food (KSNDMC) situated at Bangalore. security. Methodology The prediction of rainfall at a particular place and time can be made by studying the The procedures adopted for this work can be behavior of rainfall of that place over several summarized in the following steps: years during the past. This behavior is best studied by fitting a suitable distribution to the Statistics of annual and seasonal period time series data on the rainfall (Kainth 1996). rainfall The rainfall is predicted with the help of the probability estimates. Probability and Using the sample data (i=1, 2…, n) the basic frequency analysis of rainfall data enables to statistical descriptors of the annual rainfall determine the expected rainfall at different series, the mean, standard deviation, probability level (Mishra et al., 2013). coefficient of variation (CV), skewness, kurtosis, minimum and maximum values, The probability distributions are used in have been estimated for each taluk of Kodagu different fields of science such as engineering, district.

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Distribution Fitting (0.05) level of significance for the selection of the best fit distribution. The hypothesis under The annual average and seasonal period the GOF test is: rainfall data for each of the 3 taluk of Kodagu district are fitted to the selected 26 continuous H0: MMR data follow the specified probability distributions as presented in table 1. distribution, H1: MMR data does not follow the specified Testing the goodness of fit distribution.

The goodness-of-fit tests namely, The best fitted distribution is selected based Kolmogorov Smirnov test, Anderson-Darling on the minimum error produced, which is test and Chi-Squared test were used at α evaluated by the following techniques:

Table.1 Description for continuous probability distribution

Sl Distributio Probability Distribution Function f(x) Range/values Parameters No n  1 1 Burr (3P) x ,  ,   0 ,as hape ak  0 x     scale fx()    1 x  1    1 2 Burr (4P) x  ,as hape ak  x      scale fx()    1   location xy  1    1 3 Dagum x ,as hape (3P) ak    scale    1 x  1    1 4 Dagum x  ,as hape (4P) ak    scale    1   location xy  1   5 Fatigue xx// 1 x  , , 0 as hape Life .   0 x     scale 2 (x y )    x

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6 Fatigue , , 0 as hape

Life (3P) ()//()xx      1 x   x     scale .   2 (x y )    x    location 7 Gamma ()x  1 , , 0 as hape

(2P) f( x ) exp( x / ) () 0 x     scale 8 Gamma ()x   1 , , 0

(3P) f( x ) exp(  ( x  ) / ) ()  x  

9 1 Gen. 1 1 1 , , 0  exp( (1 zz ) )(1  )  0 Extreme    x   fx( )  1 Value exp(zz  exp(  ))  0   Gen. k 1 ,  ,   0 k, a s hape kx x k Gamma exp( ( ) ) 0 x     scale 10 k () 

11 Gen. kx() k 1 ,as hape f( x ) exp(  (( x  ) )k ) Gamma k ()  x     scale (4P)   location

12 Gumbel 1  x     scale f( x ) exp(  z  exp(  z )) Max    location Where, x   fx() 

13 Inv. 2 ,0   shape  ()x   Gaussian 32exp  22xx 0 x     location

14 2   shape Inv.  ()x    ,0 Gaussian exp  2 (xx  )32 2  (  )  x   ,  location (3P) 

15 2 Johnson  1 z x     ,  shape fx( ) exp   ln SB  2zz (1 ) 21 z   scale   location 16 2 Log- 1  Logistic  xx     1        

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17 2 Log- 1  , , 0 as hape Logistic xx       1    x     scale (3P)          location

18 Log-  1   0 1 ln(XX )   ln( )  fx( )  exp   Pearson 3 X ()         0 y 00xe    exy    0

19 Log- 2   0   shape 1 ln(x   Normal exp         scale 2  0 x   x2

20 Log- 2   0   shape 1 ln(x  ) Normal exp         scale (3P) 2   x   (x  )  2    location

21 Pearson 5 exp( x ) , , 0 as hape

(  )(x  ) 1 0 x     scale

22 Pearson 5 exp( (x )) , , 0

(3P) (  )((x  )  ) 1  x  

23 1 1 Pearson 6 ()x  ,  ,   0 12,  shape fx() 12 12   scale Bx( 12 ,  )(1  ) 0 x  

24 1 1 Pearson 6 ((x  ) ) 12,  ,   0 12,  shape fx() (4P) 12   scale Bx( 12 ,  )(1 (  )  )  x     location

25 Weibull 1 , , 0  xx    fx( ) exp       0 x        

26 Weibull 1 xx      (3P) fx( ) exp            

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Kolmogorov-Smirnov test Where Oiis the observed frequency for bin i, 2 and Eiis the expected (theoretical) x Is used to decide if a sample (x1, x2,xn ) with frequency for bin I calculated by Ei=F(X2) - CDF F(x) comes from a hypothesized F(X1), F is the CDF of the probability continuous distribution. The Kolmogorov- distribution being tested, X1 and X2 limits for Smirnov statistic (D) is based on the largest bin i. vertical difference between the theoretical CDF and the empirical (observed) CDF and is Based on Kolmogorov-Smirnov, Anderson- given by Darling and Chi-squared GOF test statistic ii1 values, 3 different rankings have been given Dmax F ( xii )  ,  F ( x ) to each of the distributions for all the taluk. 1innn No rank is given to a distribution when the

concerned test fails to fit the data. Results of A large difference indicates an inconsistency the GOF tests for all the districts are depicted between the observed data and the statistical in Table 3 to 4. model.

Identification of best fitted probability Results and Discussion distribution

Anderson-Darling Test The three goodness of fit test mentioned

above were fitted to the maximum rainfall The Anderson-Darling test it was introduced data treating different data set. The test by Anderson and Darling (1952) to place statistic of each test was computed and tested more weight or discriminating power at the at (a =0.05) level of significance. tails of the distribution. This can be important when the tails of the selected theoretical Accordingly, the ranking of different distribution are of practical significance. probability distributions was marked from 1

to 26 based on minimum test statistic value. It is used to compare the fit of an observed The distribution holding the first rank was CDF to an expected CDF. This test gives selected for all the three tests independently. more weight to the tails than the Kolmogorov The assessments of all the probability -Smirnov test. The test statistic (A2), is distribution were made on the bases of total defined as n test score obtained by combining the entire 2 1 A  n (2 i  1) ln F ( xi )  ln(1  F ( x n i 1 )) three tests. n i1 Maximum score 26 was awarded to rank first Chi-Squared Test probability distribution to the data based on the test statistic and further less score was The Chi-Squared test is used to determine if a awarded to the distribution having rank more sample comes from a population with a than 1, that is 2 to 26 and in some case where specific distribution. The Chi-Squared the distribution was not fit it was scored 0. statistic is defined as Thus, the total score of the entire three tests were summarized to identify the best fit n ()OE 2  2  ii distribution on the bases of highest score  E obtained (Sharma, 2010). i1 i

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Descriptive statistics The value of kurtosis ranges from -0.708 to 4.972. Virajpet taluk has the highest value of Mean, Minimum, Maximum, Range, Standard kurtosis implying the possibility of a deviation (SD), coefficient of variance, distribution having a distinct peak near to the skewness and Kurtosis are the descriptive mean with a heavy tail. statistics for annual and seasonal period rainfall of three taluk that are summarized in Somwarpet taluk has the smallest negative table 2 and 3. value of kurtosis which indicates that the distribution is probably characterized with a Over a span of 61 years among the three relatively flat peak near to the mean and taluks of Kodagu district, Virajpet taluk which is too flat to be normal. showed the highest range value of rainfall. Madikeri taluk received a highest mean of From table 3, we can say that Virajpet taluk 3293.094 mm while Somwarpet taluk showed the highest range value of rainfall. received a lowest mean of 2154.980 mm. Highest mean of 2771.874 mm is observed during the S-W monsoon period of Madikeri High SD (713.174 mm) is observed for taluk and the lowest mean of 208.606 mm for Madikeri taluk imply that there is large Somwarpet during Pre-monsoon period. High variation in average annual rainfall while less SD of 670.568 mm is shown during S-W variation is observed for Somwarpet taluk monsoon of Madikeri taluk imply that there is with less SD (523.763 mm). large variation in average annual rainfall while less variation is observed during pre- CV indicates the irregularities in the average monsoon period of Somwarpet taluk with SD annual rainfall. Among the three taluks, of 100.761 mm. Madikeri with less CV (21.70%) showed more consistence while Virajpet with high Pre-monsoon period of Madikeri with less CV CV (25.20%) showed relatively inconsistent. of 24. 2% showed more consistence while Skewness measures the asymmetry of a pre-monsoon period of Virajpet with high CV distribution around the mean. For all taluks of 0.523 shows inconsistency in relative the skewness is positively skewed indicating terms. For all taluks the skewness is positive. that average annual rainfall is positively The value of kurtosis ranges from -0.313 for skewed. S-W of Somwarpet to 3.136 for Post- monsoon of Virajpet.

Table.2 Descriptive statistics of annual average period for three taluks of Kodagu

Taluk Name Min Max Range Mean SD CV (%) Skewness Kurtosis

Madikeri 1929.800 5829.200 3899.400 3293.094 713.174 21.70 0.854 1.832 Somwarpet 1290.400 3215.200 1924.800 2154.980 523.763 24.30 0.249 -0.708 Virajpet 1251.200 5175.300 3924.040 2455.737 617.761 25.20 1.504 4.972

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Table.3 Descriptive statistics of monsoon period for three taluks of Kodagu

Taluk Monsoon Min Max Range Mean SD CV (%) Skewness Kurtosis Season Madikeri Pre 62.600 623.300 560.700 244.939 126.373 51.60 1.299 2.102 S-W 1462.900 4888.800 3425.900 2771.874 670.568 24.20 0.797 0.911 Post 94.400 618.000 523.600 276.281 116.374 42.10 0.912 0.612 Somwarpet Pre 86.00 589.90 503.90 208.606 100.761 48.30 1.360 2.342 S-W 875.90 2907.30 2031.40 1710.932 473.392 27.70 0.497 -0.313 Post 37.50 559.40 521.90 235.441 105.570 44.80 0.570 0.424 Virajpet Pre 66.50 691.80 625.30 252.783 132.081 52.30 1.401 2.148 S-W 1037.97 4247.90 3209.93 1936.535 578.085 29.90 1.308 3.136 Post 72.00 614.80 542.80 266.420 124.931 46.90 0.905 0.717

Table.4 Score wise best fitted probability distribution with parameter estimates for average annual rainfall of three taluks of Kodagu district

SLNo. Taluk Name of Distribution Total Score Distribution Parameter Estimates 1 Madikeri Log Logistic (3P) 75  =8.486,  =3195.000,  =28.949 2 Somwarpet Log Logistic 77 =6.728, =2076.700 3 Virajpet Gumbel Max 71  =481.600,  =2177.700

Table.5 Score wise best fitted probability distribution with parameter estimates for rainfall of monsoon period of three taluks of Kodagu district

Taluk Season Name of Distribution Total Distribution Parameter Estimates Score Madikeri Pre monsoon Dagum 78  =0.662, =4.150, =259.020

South-west Log Logistic (3P) 78 =6.561, =2322.200, =362.720 monsoon Post monsoon Log Logistic (3P) 75 =4.408 =266.140 =-10.466

Somwarpet Pre monsoon Inv. Gaussian 72  =894.130,  =208.610

South-west Gen Gamma 72 =1.006, =13.272, =130.980 monsoon Post monsoon Gamma (3P) 70 =9.704, =33.796, =-92.498

Virajpet Pre monsoon Pearson 6 59  1=13.634,  2=6.454, =101.440

South-west Pearson 5 (3P) 66 =13.609, =24600.000, =-14.481 monsoon Post monsoon Log-Logistic (3P) 78 =4.533, =297.110, =- 51.978

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It is observed from table 4 that Log-Logistic Bangalore, Karnataka (India). p 1-4,21. 3P distribution comes best fit for Madikeri Ghosh, S., Roy, M. K. and Biswas, S.C. 2016. taluk while Log-Logistic and Gumbel max Determination of the best fit Probability distribution are found to be more suitable for Distribution for Monthly Rainfall Data Somwarpet and Virajpet taluk respectively. In in Bangladesh. American J. of table 5 it is observed that Log-logistic (3P) is Mathematics and Statistics., 66(4):170- found to be most fitted distribution for S-W 174. monsoon period of Madikeri, post monsoon Kainth, G. S. 1996. Weather and Supply period of Madikeri and Virajpet taluks Behaviour in Agriculture: An respectively. Econometric Approach. Daya Books. Mandal, K.G., Padhi, J., Kumar, A., Ghosh, While Dagum, Inv. Gaussian, Gen. Gamma, S., Panda, D.K., Mohanty, R.K., Gamma, Pearson 6 and Pearson 5 (3P) Raychaudhuri, M. 2014.Analyses of distributions were found to be most suitable rainfall using probability distribution for Pre-monsoon period (Madikeri), Pre- and Markovchain models for crop monsoon period (Somwarpet), S-W monsoon planning in Daspalla region in Odisha, period (Somwarpet), Post-monsoon period India. J. Theor Appl. Climatol.,121(3- (Somwarpet), Pre-monsoon period (Virajpet) 4):517-528. and S-W monsoon period (Virajpet) Mishra, P. K., Khare, D., Mondal, A., Kundu, respectively. S. and Shukla, R. 2013.“Statistical and Probability Analysis of Rainfall for References Crop Planning in A Canal Command”. Agriculture for Sustainable Anonymous. 2018.Annual and Seasonal Development, 1(1):45-52. Rainfall Pattern and Area Coverage Sukrutha, A., Dyuthi, S.R. And Desai, S. during Kharif and Rabi seasons of 2017, 2018.Multimodel response assessment Directorate of Economics and statistics, for monthly rainfall distribution in some special report No. DES/ 10 /2018. selected cities using best-fit probability Bhavyashree, S. and Bhattacharyya, B. 2018. as a tool. Open access J. Applied water Fitting Probability Distribution for Sci., 8(5):145 rainfall Analysis of Karnataka, India. Yue, S., and Hashino, M. 2007. Probability Int. J. Curr. Microbiol. and App. Sci., distribution of annual, seasonal and 7(3):2319-7706. monthly precipitation in Japan. Deepthi, K.A. 2015. Analysis of Temporal Hydrological sci. J. des. Sci. and Spatial rainfall of Kodagu District. Hydrologiques., 52(5):863-877. Ms.c. thesis, Univ. Agri. Sci.,

How to cite this article:

Shreyas. R, D. Punith, L. Bhagirathi, Anantha Krishna and Devagiri. G. M. 2020. Exploring Different Probability Distributions for Rainfall Data of Kodagu - An Assisting Approach for Food Security. Int.J.Curr.Microbiol.App.Sci. 9(02): 2972-2980. doi: https://doi.org/10.20546/ijcmas.2020.902.339

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