Multimodel Response Assessment for Monthly Rainfall Distribution in Some Selected Indian Cities Using Best Fit Probability as a Tool

We carry out a study of the statistical distribution of rainfall precipitation data for 20 cites in India. We have determined the best-fit probability distribution for these cities from the monthly precipitation data spanning 100 years of observations from 1901 to 2002. To fit the observed data, we considered 10 different distributions. The efficacy of the fits for these distributions was evaluated using four empirical non-parametric goodness-of-fit tests namely Kolmogorov-Smirnov, Anderson-Darling, Chi-Square, Akaike information criterion, and Bayesian Information criterion. Finally, the best-fit distribution using each of these tests were reported, by combining the results from the model comparison tests. We then find that for most of the cities, Generalized Extreme-Value Distribution or Inverse Gaussian Distribution most adequately fits the observed data.


INTRODUCTION
Establishing a probability distribution that provides a good fit to the monthly average precipitation has long been a topic of interest in the fields of hydrology, meteorology, agriculture. The knowledge of precipitation at a given location is an important prerequisite for agricultural planning 1 arXiv:1708.03144v2 [stat.AP] 25 Jan 2018 and management. Rainfall is the main source of precipitation. Studies of precipitation provide us knowledge about rainfall. For rain-fed agriculture, rainfall is the single most important agrometeorological variable influencing crop production. In the absence of reliable physically based seasonal forecasts, crop management decisions and planning have to rely on statistical assessment based on the analysis of historical precipitation records. In fact it has been shown by Fis (1925) that the statistical distribution of rainfall is more important than the total amount of rainfall for the yield of crops. Therefore, detailed statistical studies of rainfall data for a variety of countries have been carried out for more than 70 years along with fits to multiple probability distribution (Ghosh et al. 2016;Sharma and Singh 2010;Nguyen et al. 2002). We only mention a few selected works amongst these. More details can be found in the above works and references therein. We first enumerate these studies for various stations in India and then briefly list similar studies outside India. Mooley and Appa Rao (1970) first carried out a detailed statistical analysis of the rainfall distribution during southwest and northeast monsoon seasons at selected stations in India with deficient rainfall and found that the Gamma distribution provides the best fit. The Gamma function was also used to fit the daily precipitation data of Coimbatore (Kulandaivelu 1984) and for the frequency distribution of consecutive days of peak rainfall in Banswara, Rajasthan (Bhakar et al. 2006). Sharda and Das (2005) found that the Weibull distribution provides the best fit for rainfall distribution near Dehradun. In addition to the total rainfall, Stephenson et al. (1999) showed that the outliers in the rainfall distribution for the summers of 1986 to 1989 throughout India can be well fitted by the gamma and Weibull distributions. Deka et al. (2009) found that the logistic distribution is the optimum distribution for the annual rainfall distribution for seven districts in north-East India. Sharma and Singh (2010) found based on daily rainfall data for Pantnagar spanning 37 years, that the lognormal and gamma distribution provide the best fit probability distribution for the annual and monsoon months, whereas the Generalized extreme value provides the best fit after considering only the weekly data. Most recently, Kumar et al. (2017) analyzed the statistical distribution of rainfall in Uttarakhand, India and found that the Weibull distribution performed the best, while the second best distribution was Chi squared (2P) and log-Pearson. However, one caveat with some of the above studies is that only a handful of distributions were considered for fitting and sometimes no detailed model comparison tests or statistical tests were done to find the most adequate distribution.
A large number of statistical studies have similarly been done for rainfall precipitation data for stations outside India. In Costa Rica, normal distribution provided the best fit to the annual rainfall distribution (Waylen et al. 1996). A Log-Pearson type III distribution was used for fitting the rainfall data in Texas (Salami 2004) and China (Lee 2005). On the other hand, a generalized extreme value distribution has been used for Louisiana (Naghavi and Yu 1995) and Ontario (Pilon et al. 1991).
Gamma distribution provided the best fit for rainfall data in Saudi Arabia (Abdullah and Al-Mazroui 1998), Sudan (Mohamed and Ibrahim 2015) and Libya (near the arid regions) (ŞEN and Eljadid 1999). Mahdavi et al. (2010) studied the rainfall statistics for 65 stations in the Mazandaran and Golestan provinces in Iran and found that the Pearson and log-Pearson distribution provide the best fits to the data. Nadarajah and Choi (2007) found that Gumbel distribution provides the most reasonable fit to the data in South Korea. Ghosh et al. (2016) fitted the rainfall data from five stations in Bangladesh to six probability distributions and used three different tests to select the distribution with the maximum efficacy. They found that the extreme value distribution provides the best fit to the Chittagong monthly rainfall data during the rainy season, whereas for Dhaka, the gamma distribution provides a better fit. Most recently, Silva and Peiris (2017) found that the three parameter Weibull distribution is the best fit describing 56 years worth of rainfall data in the city of Colombo in Sri Lanka. Therefore, we can see from these whole slew of studies, that no single distribution can accurately describe the rainfall distribution. The selection depends on the characteristics of available rainfall data as well as the statistical tools used for model selection.
The main objective of the current study is to complement the above studies and to determine the best fit probability distribution for the monthly average precipitation data of 20 selected stations throughout India. The best fit probability distribution was evaluated on the basis of several goodness of fit tests. We first use the same goodness of fit tests as in Ghosh et al. (2016), and furthermore apply two additional ones from information theory, viz. Akaike and Bayesian Information Criterion for each of these cities. We then combine the results from each of these goodness of fit tests to find the most optimum distribution for each city.
The precipitation-based information generated by this study is expected to be of considerable agronomic importance for the efficient planning and management of rain fed cotton based cropping system. Apart from agriculture point of view, the current study also finds its importance in the estimation of extreme precipitation for design purposes, prediction of return periods of rainfall, assessment of rarity of observed precipitation, comparison of methods to estimate design precipitation which is useful for planners.
The outline of this paper is as follows. The dataset used for this analysis is described in Sect. 2.

DATASET
The dataset employed here for our study spans a 100 year period from 1901 to 2002, and is collected by the Indian Meteorological Department. This data can be downloaded from http: //www.indiawaterportal.org/met_data/. We used 20 stations for our study. The stations used for this study are Gandhinagar, Guntur, Hyderabad, Jaipur, Kohima, Kurnool, Patna, Aizwal, Bhopal, Ahmednagar, Cuttack, Chennai, Bangalore, Patna, Amritsar, Guntur, Lucknow, Kurnool, Jammu, Delhi, and Panipat. The location of these stations on the map of India is shown in Fig. 1.
We briefly describe some of the salient features of these stations below: Gandhinagar Gandhinagar is the Capital of Gujarat and located in the western part of India. Its

METHODOLOGY
The probability distributions considered for fitting the rainfall data are gamma, Fisher, Inverse Gaussian, Normal, Student−t, LogNormal, Generalized Extreme value, Weibull, and Beta. The mathematical expressions for the probability density functions of these distributions can be found in Table 1 and have been adapted from (VanderPlas et al. 2012;Ghosh et al. 2016). All of these distributions have have been previously used for similar studies. For each station, we find the best-fit parameters for each of these probability distribution using maximum-likelihood analysis. To select the best-fit distribution for a given station, we then use multiple model comparison techniques to rank each distribution for every city. We now describe the model comparison techniques used.

Model Comparison tests
We use multiple model comparison methods to carry out hypothesis testing and select the best distribution for the precipitation data. The test is performed in order to select among the following hypotheses: H0 : The amount of monthly precipitation data follows a specified distribution.
H1 : The amount of monthly precipitation data does not follow the specified distribution. In other words, H1 is the complement of H0.
The goodness of fit tests conducted herein include non-parametric distribution-free tests such as Kolmogorov−Smirnov Test, Anderson-Darling Test, Chi−square test and information-criterion tests such as Akaike and Bayesian Information Criterion. For each of the probability distributions, we find the best-fit parameters for each of the stations using least-squares fitting and then carry out the different model comparison tests discussed below.

Kolmogorov-Smirnov Test
The Kolmogorov-Smirnov (K-S) test (Vetterling et al. 1992) is a non-parametric test used to decide if a sample comes from a population with a specific distribution. The K-S test compares the empirical distribution function (ECDF) of two samples. Given N ordered data points y 1 , y 2 , ..., y N , the ECDF is defined as where n(i) indicates the total number of points less than y i after sorting the y i in increasing order.
This is a step function that increases by 1/N for each sorted data point.
The K-S test is based on the maximum distance (or supremum) between the empirical distribution function and the normal cumulative distributive function. An attractive feature of this test is that the distribution of the K-S test statistic itself does not depend on the statistics of the parent distribution from which the samples are drawn. Some limitations are that it applies only to continuous distributions and tends to be more sensitive near the center of the distribution than at the tails.
The Kolmogorov-Smirnov test statistic is defined as: where F is the cumulative distribution function of the samples (or a probability distribution function) being tested. If the probability that a given value of D is very small (less than a certain critical value, which can be obtained from tables) we can reject the null hypothesis that the two samples are drawn from the same underlying distributions at a given confidence level.

Anderson-Darling Test
The where F is the cumulative distribution function of the specified distribution and y i denotes the sorted data. The test is a one-sided test and the hypothesis that the data is sampled from a specific distribution is rejected if the test statistic, A, is greater than the critical value. For a given distribution, the Anderson-Darling statistic may be multiplied by a constant (depending on the sample size, n). These constants have been tabulated by Stephens (1974).

Chi-Square Test
The chi-square test (Cochran 1952) is used to test if a sample of data is obtained from a population with a specific distribution. An attractive feature of the chi-square goodness-of-fit test is that it can be applied to any univariate distribution for which you can calculate the cumulative distribution function. The chi-square goodness-of-fit test is usually applied to binned data. The chi-square goodness-of-fit test can be applied to discrete distributions such as the binomial and the Poisson distributions. The Kolmogorov-Smirnov and Anderson-Darling tests are restricted to continuous distributions. For the chi-square goodness-of-fit computation, the data are divided into k bins and the test statistic is defined as follows: where O i is the observed frequency for bin i and E i is the expected frequency for bin i. The expected frequency is calculated by where F is the cumulative distribution function for the distribution being tested, Y u is the upper limit for class i, Y l is the lower limit for class i, and N is the sample size.
This test is sensitive to the choice of bins. There is no optimal choice for the bin width (since the optimal bin width depends on the distribution). For our analysis, since there were a total of 1224 data points, we have chosen 100 bins, so that there were sufficient data points in each bin. For the chi-square approximation to be valid, the expected frequency of events in each bin should be at least five. The test statistic follows, approximately, a chi-square distribution with (k − c) degrees of freedom where k is the number of non-empty cells and c is the number of estimated parameters (including location, scale, and shape parameters) for the distribution + 1. Therefore, the hypothesis that the data are from a population with the specified distribution is rejected if: where χ 1−α,k−c 2 is the chi-square critical value with k − c degrees of freedom and significance level α.

AIC and BIC
The Akaike Information Criterion (AIC) (Liddle 2004;Kulkarni and Desai 2017) is a way of selecting a model from a set of models. It can be derived by an approximate minimization of the Kullback-Leibler distance between the model and the truth. It is based on information theory, but a heuristic way to think about it is as a criterion that seeks a model that has a good fit to the truth with very few parameters.
It is defined as (Liddle 2004): where L is the likelihood which denotes the probability of the data given a model, and K is the number of free parameters in the model. AIC scores are often shown as ∆AIC scores, or difference between the best model (smallest AIC) and each model (so the best model has a ∆AIC of zero).
The bias-corrected information criterion, often called AICc, takes into account the finite sample size by, essentially, increasing the relative penalty for model complexity with small data sets. It is defined as (Kulkarni and Desai 2017): where L is the likelihood and N is the sample size. For this study we have used AICc for evaluating model efficiacy.
Bayesian information criterion (BIC) is also an alternative way of selecting a model from a set of models. It is an approximation to Bayes factor between two models. It is given by (Liddle 2004): When comparing the BIC values for two models, the model with the smaller BIC value is considered better. In general, BIC penalizes models with more parameters more than AICc does.

RESULTS AND DISCUSSION
Some basic statistics for the amount of monthly precipitation data for the above mentioned stations are summarized in Table 2, where the minimum, maximum, mean, standard deviation (SD), coefficient of variation (CV), skewness, and kurtosis are shown. The monthly rainfall dataset indicates that the monthly rainfall was strongly positively skewed for Gandhinagar, Jaipur, Amritsar, Delhi, and Panipat stations. Aizwal, Kohima, and Cuttack show negative values of kurtosis.
The distributions listed above are fitted for the each of the selected locations. For brevity, in this manuscript, we show the plots for only four cities. These can be found in Figures [2-5], which illustrate the fitted distribution for Kurnool, Hyderabad, Jammu, and Patna. Similar plots for the remaining stations have been uploaded on a google drive, whose link is provided at the end of this manuscript.
The test statistics for K-S test (D), Anderson-Darling Test (A 2 ), Chi-square test ( χ 2 ), AICc, and BIC were computed for the ten probability distributions. The AICc and BIC values for each of these 10 distributions and 20 cities can be found on the google drive, which documents this analysis. The probability distribution that fits a given data the best (using the largest p-value) according to each of the above criterion is shown in Table 3.
For each station, we ranked all the probability distribution functions using each of the four model comparison techniques in decreasing order of its p-value. The best fit distribution amongst these, for each city was found after summing these ranks and choosing the function with the smallest summed rank. A similar technique was also used in Sharma and Singh (2010) to find the best distribution, which fits the rainfall data using multiple model comparison techniques. The best fit distribution for each station using this ranking technique is shown in Table 4 Table 5.
Our results from each of the model comparison tests are summarized as follows: • Using K-S test (D), we find that the Fisher distribution provides a good fit to the monthly precipitation data for all cities except Kohima and Aizawl. For these cities, Weibull distribution provide the best fit.
• Using Anderson-Darling Test (A 2 ), it is observed that the Fisher distribution is the best fit for all the cities except (again) for Kohima and Aizawl, for which the Beta distribution gives the best fit for both the cities.
• Using Chi-square test ( χ 2 ), there is no one distribution which consistently provides the best fit for most of the cities. Inverse Gaussian is the optimum fit for seven cities, whereas Weibull and Generalized extreme for three cities, Beta and Fisher for two cities each. The locations of the corresponding cities can be found in Table 3.
• Using AICc, it is observed that the Fisher distribution provides best distribution for about 16 cities. The exceptions are again Kohima and Aizawl, for which Weibull is the most appropriate distribution. Generalized extreme value distribution provides the best fit for Gandhinagar, whereas Students t-distribution provides the best fit for Ahmednagar.
• For BIC, we find that the beta distribution provides best distribution for all districts except Gandhinagar. Student-t distribution provides best fit for Gandhinagar.
If we then determine the best distribution from a combination of the above model comparison techniques using the ranking technique, we find (cf. Table 4) that the generalized extreme value distribution is the most appropriate for eight cities, inverse Gaussian for nine cities, Gumbel for two cities, and gamma for one city. Therefore, although no one distribution provides the best fit for all stations, for most of them can be best fitted using either the generalized extreme value or inverse Gaussian distribution.

IMPLEMENTATION
We have used the python2.7 environment. In addition, Numpy, pandas, matplotlib, scipy packages are used. Our codes to reproduce all these results can be found in http://goo.gl/ hjYn1S. These can be easily applied to statistical studies of rainfall distribution for any other station.

COMPARISON TO PREVIOUS RESULTS
A summary of some of the previous studies of rainfall distribution for various stations in India is outlined in the introductory section. An apples-to-apples comparison to these results is not straightforward, since they have not used the same model comparison techniques or considered all the 10 distributions which we have used. Moreover, the dataset and duration they have used is also different. Nevertheless, we compare and contrast the salient features of our conclusions with the previous results.
Among the previous studies, Sharma and Singh (2010) have also found that Generalized extreme value distribution fits the weekly rainfall data for Pantnagar. We also find that this distribution provides the best fit for eight cities. The best-fit distribution which we found for Aizawl agrees with the results from (Mooley and Appa Rao 1970;Kulandaivelu 1984;Bhakar et al. 2006).
None of the previous studies have found the Inverse Gaussian or the Gumbel distribution to be an adequate fit to the rainfall data. However, this could be because these two distributions were not fitted to the observed data in any of the previous studies. Inverse Gaussian and the Gumbel distribution have only recently been considered by Ghosh et al. (2016) and Nadarajah and Choi (2007) for fitting the rainfall data in Bangladesh and Korea respectively. We hope our results spur future studies to consider these distributions for fitting rainfall data in India.

CONCLUSIONS
We carried out a systematic study to identify the best fit probability distribution for the monthly precipitation data at twenty selected stations distributed uniformly throughout India. The data showed that the monthly minimum and maximum precipitation at any time at any station ranged from 0 to 802 mm, which obviously indicates a large dynamic range. So identifying the best parametric distribution for the monthly precipitation data could have a wide range of applications in agriculture, hydrology, engineering design, and climate research.
For each station, we fit the precipitation data to 10 distributions described in Table 1. To determine the best fit among these distributions, we used five model comparison tests, such as K-S test, Anderson-Darling test, chi-square test, Akaike and Bayesian Information criterion. The results from these tests are summarized in Table 3. For each model comparison test, we ranked each distribution according to its p-value and then added the ranks from all the four tests. The best-fit distribution for each city is the one with the minimum total rank. This best-fit distribution for each city is tabulated in Table 4. We find that no one distribution can adequately describe the rainfall data for all the stations. For about nine cities, the Inverse Gaussian distribution provides the best fit, whereas Generalized extreme value can adequately fit the rainfall distribution for about eight cities. Our study is the first one, which finds the Inverse Gaussian distribution to be the optimum fit for any station. Among the remaining cities, Gumbel and Gamma distribution are the best fit for two and one city respectively.
In the hope that this work would be of interest to researchers wanting to do similar analysis and to promote transparency in data analysis, we have made our analysis codes as well as data publicly available for anyone to reproduce this results as well as to do similar analysis on other rainfall datasets. This can be found at http://goo.gl/hjYn1S

Distribution
Probability density function  Map showing location of various stations throughout India for which rainfall statistics and best-fit distributions were obtained. Each red point represents a station and next to it we show its first three letters. The full names of the cities can be found in Table 1. This plot has been made with the ggplot (Kahle and Wickham 2013) package in the R programming language.

Fig. 2.
Histogram of the monthly precipitate data at Kurnool (blue lines) along with best fit for each of the 10 probability distributions functions considered. Fig. 3. Histogram of the monthly precipitate data at Hyderabad (blue lines) along with best fit for each of the 10 probability distributions functions considered. Fig. 4. Histogram of the monthly precipitate data at Jammu (blue lines) along with best fit for each of the 10 probability distributions functions considered.

Fig. 5.
Histogram of the monthly precipitate data at Patna (blue lines) along with best fit for each of the 10 probability distributions functions considered. We note that the corresponding plots for all the remaining cities have been uploaded on the google drive