Modeling the Distribution of Annual Maximum Rainfall in Pakistan

Abstract

In this study an effort has been made to figure out the best fitting distribution to explain the annual maximum of daily rainfall for the years 1954 to 2005 of four stations in Pakistan. For this purpose Gamma, Generalized Extreme Value (GEV) and Generalized Pareto (GP) distributions are fitted to annual maximum of daily rainfall data from each station. Parameters for each distribution are estimated using the Maximum Likelihood (ML) method and Probability Weighted Moment (PWM) method. The performance of the distributions have been investigated using different goodness of fit (GOF) tests namely Chi-Square (CS), Kolmogorove-Smirnove (KS) and Anderson Darling (AD) test. Further, Probability Probability (PP) and Quantile Quantile (QQ) plots are used to confirm the goodness of fit for Gamma, GEV and GP distribution. Finally, goodness of fit tests are compared suggesting that GEV and GP distributions are found to be most appropriate distribution for describing the behavior of annual maximum of daily rainfall in Pakistan for selected stations whereas Gamma distribution is also a reasonable model for three of the four stations considered. Return levels for different years are estimated using GEV distribution and describe how they vary with the stations.

Full text

European Journal of Scientific Research
ISSN 1450-216X Vol.79 No.3 (2012), pp.418-429
© EuroJournals Publishing, Inc. 2012
http://www.europeanjournalofscientificresearch.com
Modeling the Distribution of Annual Maximum
Rainfall in Pakistan
Kamran Abbas
Department of Statistics, University of Azad Jammu & Kashmir
Muzaffarabad, Pakistan
Alamgir
Department of Statistics, University of Peshawar, Pakistan
Sajjad Ahmad Khan
Department of Statistics, Islamia College University
Peshawar, Pakistan
Dost Muhammad Khan
Corresponding Author, Department of Statistics
Islamia College University, Peshawar, Pakistan
E-mail: profalamgir@yahoo.com; dost_uop@yahoo.com
Amjad Ali
Department of Statistics, Islamia College University
Peshawar, Pakistan
Umair khalil
Department of Statistics, Islamia College University
Peshawar, Pakistan
Abstract
In this study an effort has been made to figure out the best fitting distribution to
explain the annual maximum of daily rainfall for the years 1954 to 2005 of four stations in
Pakistan. For this purpose Gamma, Generalized Extreme Value (GEV) and Generalized
Pareto (GP) distributions are fitted to annual maximum of daily rainfall data from each
station. Parameters for each distribution are estimated using the Maximum Likelihood
(ML) method and Probability Weighted Moment (PWM) method. The performance of the
distributions have been investigated using different goodness of fit (GOF) tests namely
Chi-Square (CS), Kolmogorove-Smirnove (KS) and Anderson Darling (AD) test. Further,
Probability Probability (PP) and Quantile Quantile (QQ) plots are used to confirm the
goodness of fit for Gamma, GEV and GP distribution. Finally, goodness of fit tests are
compared suggesting that GEV and GP distributions are found to be most appropriate
distribution for describing the behavior of annual maximum of daily rainfall in Pakistan for
selected stations whereas Gamma distribution is also a reasonable model for three of the
four stations considered. Return levels for different years are estimated using GEV
distribution and describe how they vary with the stations.

Modeling the Distribution of Annual Maximum Rainfall in Pakistan 419
Keywords: Annual maximum daily rainfall, Gamma distribution, Generalized Extreme
Value distribution, Generalized Pareto distribution, Goodness of fit tests.
1. Introduction
The climate of Pakistan is generally characterized by the hot summers and the cold winters. Also
Pakistan has wide variations among extremes of temperature at given locations. Pakistan has four
seasons: a cool and dry winter from December through February; a hot and dry spring from March
through May; the summer (rainy) season from June through September; and the retreating monsoon
period of October and November. The lengths of these seasons vary to some extent according to the
locations. Weather is the combination of different climate conditions such as temperature, rainfall,
pressure, humidity, sunshine, wind speed and cloudiness. Among these climate conditions, rainfall is
the most important one because heavy rainfalls could bring disaster such as flood and landslide.
Pakistan has faced so many floods. The worst of these is the summer 2010 flood which shocked all
Pakistan in general and Sindh and Khyber-Pukhtoonkhwa provinces in particular. In addition, the
shortage of rainfalls has also an effect on water management system and other economic activities.
Rainfall is the percentage refers to the likelihood of an area getting a measurable amount of rain over a
particular time period. Therefore, regionalized study according to statistical perspective is essential to
model the extreme rainfall behavior. Modeling of daily rainfall using various mathematical models has
been applied throughout the world to develop a parsimonious model about the rainfall pattern and its
characteristics. Several methods have been proposed in the literature for modeling rainfall amount. e. g.
Suhaila and Jemain (2008) studied several types of Exponential distributions to describe rainfall
behavior in Peninsular Malaysia over a multiyear period. They concluded that the mixture of two
distributions is better than single distributions for describing the daily rainfall amount. Similarly,
Nadarajah and Choi (2007) discussed GEV distribution to describe the extremes of rainfall and to
predict its future behaviour. They found that Gumbel distribution provides the most reasonable model
for four of the five locations considered. Further, different researchers throughout the world provided
the productive applications of extreme value distributions to rainfall data: see Park et al. (2001), Park
and Jung (2002) for applications in South Korea; Elnaqa and Abuzeid (1993) for application in Jordan;
Zalina et al. (2002) for application in Malaysia; Withers and Nadarajah (2000) for application in New
Zealand. Azumi et al. (2010) applied different distributions namely, Generalized Pareto (GP),
Exponential (EXP), Beta (BT) and Gamma (GM) distributions were used to model the distribution of
the hourly rainfall intensity and Kolmogorov‐Smirnov (KS), Anderson Darling (AD) and Chi‐
Squared (CS) goodness of fit tests were used to evaluate the best fit at 5% level of significance. They
concluded that the GP distribution is the most suitable model. Moreover, Deka et al. (2009) determined
the best fitting distribution to describe the annual series of maximum daily rainfall in North-East India.
Five extreme value distributions viz. Generalized Extreme Value (GEV) distribution, Generalized
Logistic (GL) distribution, Generalized Pareto (GP) distribution, Lognormal and Pearson distribution
are fitted and concluded that generalized Logistic distribution is empirically proved to be the most
appropriate distribution. On the other hand, Zin et al. (2008) compared GEV and GP distributions to
model the extreme rainfall in peninsular Malaysia and concluded that GEV and GP models are well
fitted. Husak et al. (2007) used Gamma distribution to represent monthly rainfall in Africa. They
demonstrated that feasibility of fitting cell-by-cell probability distributions to grids of monthly
interpolated, continent wide data and concluded that Gamma distribution is well suited to these
applications.
The purpose of this paper is to find the most appropriate distribution(s) for describing the
annual maxima of daily rainfall. Therefore, we intend to compare Gamma, GEV and GP distributions
by using different goodness of fit criterion such as Chi-Squared (CS), Kolmogrov Simirnov (KS) and
Anderson Darling (AD) test. Once a distribution function is assumed to be selected for study at hand, it
remains to estimate its parameters from the sample data and to test the goodness of fit. Several methods
have been proposed to estimate the parameters of candidate distributions, e.g. Singh (1987) compared
Principle of Maximum Entropy (POME), Method of Moments (MM) and Maximum Likelihood

420 Kamran Abbas, Alamgir, Sajjad Ahmad Khan,
Dost Muhammad Khan, Amjad Ali and Umair khalil
Estimation (MLE) for the precipitation data used. Similary, Moisello (2007) used Partial Probability
Weighted Moments (PPWM) and Probability Weighted Moments (PWM) for estimating the
hydrologcal extremes. Among the various estimation methods, Maximum Likelihood (ML) and
Probability Weighted Moment (PWM) methods are adopted in this study for parameters estimation of
the candidate distributions.
2. Materials and Methods
2.1. About the Data
In this study annual maximum daily rainfall (mm) data for the years from 1954 to 2005 for four
stations in Pakistan viz Islamabad, Murree, Lahore and Sialkot were considered. The geographical
positions of four stations are shown in Figure1. The annual maximum daily rainfall series is collected
from Meteorological department Islamabad.
Figure 1: Locations of the Rain Sampling station.
2.2. Methodology
In order to describe the pattern of maximum rainfall at a given station, it is vital to identify the
distribution (s), which adequately fit the data. In this study two parameter Gamma, three parameter
GEV and GP distributions were used to model the distribution of the rainfall amount along with
different goodness of fit statistic namely, Chi-Squared (CS), Kolmogrove Simirnov (KS) and Anderson
Darling (AD) test. The Probability Density Function (PDF), Cumulative Distribution Function (CDF)
and Quantile function
X(T) for the mentioned models are given below.
2.3. GEV Distribution
The GEV distribution with continuous location parameter ( ), continuous scale parameter ( ) and
continuous shape parameter ( ) has the PDF, CDF and X(T) given by:
(1)
for
(2)
for

Modeling the Distribution of Annual Maximum Rainfall in Pakistan 421
(3)
2.4. ML and PWM Estimators for GEV Distribution
The log-likelihood function for GEV distribution is
Where , The MLE of and can be identified by solving the
following equations. Therefore
(4)
(5)
(6)
The Newton Raphson method was used to solve the above equations following Hosking (1985)
and Macleod (1989). Similarly, the PWM estimators of GEV distribution for have been proposed
by Hosking et al. (1985) as
(7)
(8)
To obtain the value of we must solve the equation
(9)
The scale and location parameters can be estimated as
2.5. Gamma Distribution
The PDF and CDF of Gamma distribution having shape parameter , and scale parameter are
given by:
x>0, α, β >0 (10)
(11)
where is the Incomplete Gamma Function.
2.6. ML and PWM Estimators for Gamma Distribution
The Maximum likelihood estimates of and are
(12)
,
(13)

422 Kamran Abbas, Alamgir, Sajjad Ahmad Khan,
Dost Muhammad Khan, Amjad Ali and Umair khalil
Similarly, the probability weighted moment estimates of and are
(14)
2.7. GP Distribution
The PDF and CDF of GP distribution with shape parameter (a), scale parameter (b) and location
parameter (c) are given by:
(15)
(16)
2.8. ML and PWM Estimators for GP Distribution
The Maximum likelihood estimates (MLE) for shape (a), scale (b) and location (c) parameter can be
expressed as
(17)
(18)
Obviously, there are no explicit expressions for a, b and c. An MLE cannot be obtained for the
parameter c, because the likelihood function is unbounded with respect to c, the lower bound of the
random variable X. In this study all events beyond a given threshold are selected (Lang et al. 1999,
Rosberg and Madsen 2004) and then find the maximum likelihood estimates for shape (a) and scale (b)
parameters. Similarly, the PWM estimators for the GP distribution (Hosking and Wallis, 1987) are
given as
(19)
(20)
(21)
Where the rth probability weighted moment is
(22)

Modeling the Distribution of Annual Maximum Rainfall in Pakistan 423
3. Goodness of Fit Tests
Three goodness of fit tests were applied at 5 level of significance to verify the goodness of fit for the
fitted distributions in which X represent the rainfall random variable and ‘n’ the sample size . The tests
are as follows:
3.1. Chi-Square (CS) Test
This test is used to determine whether the sample comes from a specific distribution. The Chi‐Square
test statistic is defined as
,
Where is the observed frequency in each category, and is the expected (theoretical)
frequency in the corresponding category calculated by
,
Where F is the CDF of the probability distribution being tested and , are the lower and
upper limits for category . Where i runs from 1,..,k and k represent the number of cells.
3.2. Kolmogorov-Smirnov (KS) Test
The Kolmogorov-smirnov test is a nonparametric alternative test of the Chi-square goodness of fit test
and is used to confirm the sample under consideration selected from a reference or hypothesized
distribution or to compare two samples come from identical distribution. The KS test statistics is
calculated from the largest vertical difference in absolute value between the theoretical and the
empirical cumulative distribution functions. Assume that we have a random sample from
some distribution with CDF . The KS test statistic where
and .
3.3. Anderson Darling (AD) Test
This test is generally used to test the fit of an observed cumulative distribution to a theoretical specified
cumulative distribution. Unlike KS test, it gives more weight to the tails. The test statistic for the test is
as follows:
4. Results and Discussion
Table 1 presents the latitude and the longitude of four stations and basic rainfall amount statistics of the
related data sets. Among the four stations, Lahore station showed the least annual maximum daily
rainfall whereas Murree station showed the highest annual maximum daily rainfall. It is also
concluding that coefficient of variation (CV) for Murree and Islamabad is found to be lowest as
compared to other stations. This may also indicate that the consistency of extreme rainfall in those
stations as compared to other stations. In addition, Figure 2 depicts that how the annual maximum daily
rainfall has varied from 1954 to 2005 for the four stations.
Table 1: The Latitude, Longitude and summary statistics for Stations
Stations Latitude Longitude Mean SD Min. Max.
Islamabad 265.2 147.0321 33.0 743.3
Murree 341.5 130.1263 35.3 704.3
Lahore 79.73 97.5527 15.40 523.20
Sialkot 101.0 83.8407 10.10 311.90

424 Kamran Abbas, Alamgir, Sajjad Ahmad Khan,
Dost Muhammad Khan, Amjad Ali and Umair khalil
F
i
gure 2:
Annua
l
Max
i
mum
Da
il
y
Ra
i
nfa
ll
recoded
i
n
(a)
Is
l
amabad,
(b) Murree, (c)
Lahore
and
(d)
S
i
a
l
kot
for
1954
to
2005
Year
A nnual Maximum Daily Rainfall in Murree (mm)
2010200019901980197019601950
700
600
500
400
300
200
100
0
(b)
Year
A nnual Maximum Daily Rainfall in Islamabad (mm)
2010200019901980197019601950
800
700
600
500
400
300
200
100
0
(a)
Year
Annual Maximum Daily Rainfall in Lahore (mm)
2010200019901980197019601950
500
400
300
200
100
0
(c) Year
Annual Maximum Daily Rainfall in Sialkot (mm)
2010200019901980197019601950
350
300
250
200
150
100
50
0
(d)
4.1. Fitting Distributions
GEV distribution is fitted to data and parameters are estimated by using Maximum Likelihood (ML)
and Probability Weighted Moment (PWM) methods and results are presented in Table 2 for
comparison purpose. It is observed that ML and PWM estimates of location ( ), scale ( ) and shape
( ) parameters for Islamabad, Murree and Lahore stations are not significantly different but
considerably different from Sialkot station. Similarly, PWM and ML estimators for Gamma
distribution are presented in Table 3. We observed that ML and PWM estimates of shape parameter ( )
are notably same for four stations but estimates of scale parameter ( ) are significantly different.
Moreover, PWM and ML estimators for GP distribution are estimated and presented in Table 4. We
found that ML and PWM estimates of shape parameter (a) do not differ significantly but the estimates
of scale parameter (b) are significantly different, whereas ML estimate for location parameter (c) is
chosen as the threshold value i.e. the lower bound of the observed random variable.
Table 2:
Estimation of parameters for GEV distribution
ML PWM
Stations
Islamabad 199.80 111.32 0.01 197.52 108.47 0.04
Murree 291.29 122.89 -0.19 290.62 121.25 -0.19
Lahore 36.89 24.13 0.69 38.70 27.43 0.49
Sialkot 45.09 41.78 0.68 59.16 60.55 0.10

Modeling the Distribution of Annual Maximum Rainfall in Pakistan 425
Table 3:
Estimation of parameters for Gamma distribution
ML PWM
Stations
Islamabad 3.23 82.19 3.38 78.66
Murree 5.60 61.0 6.76 50.47
Lahore 1.44 55.59 1.05 75.78
Sialkot 1.32 76.72 1.23 82.58
Table 4:
Estimation of parameters for GP distribution
ML PWM
Stations
Islamabad -0.41 321.0 -0.34 244.87 81.8
Murree -0.73 493.97 -0.79 362.49 138.75
Lahore 0.32 44.89 0.37 39.67 16.23
Sialkot ‐0.29 120.90 ‐0.23 127.92 -3.02
4.2. Comparison of Probabilistic Models
To compare the probabilistic models, several goodness of fit: Kolmogorov-Smirnov test (1941),
Simirnov (1944), Anderson Darling (AD) (1952) and pearson’s chi-square (CS) test (1983) were used
to quantify the best model. The P-values of this goodness of fit statistics have been calculated and
presented in Table 5. P-values of KS test are calculated for GEV, Gamma and GP distributions for the
parameters estimation of PWM and ML methods. Results of KS test clearly indicated that GEV and
Gamma distributions are well fitted but the p-values of KS test are non-significant (Murree) for
parameters estimation of GP distribution of ML and PWM methods. Further, p-values of AD and CS
test for GEV and GP distributions are significant at 5% level of significance which may indicate that
GEV and GP distributions are most appropriate for describing the extreme rainfalls in Pakistan. In
addition, AD and CS tests are also applied when the parameters of the Gamma distribution are
estimated by ML method. P-values of AD and CS tests for Islamabad, Murree and Sialkot stations are
found to be significant at 5% level of significance but non-significant for Lahore station suggesting
that the Gamma distribution is also reasonable model for three of the four stations. In this paper that
distribution is considered most suitable when at least twice out of the three goodness of fit tests best
described the extreme rainfalls.
Finally, Probability Probability (PP) and Quantile Quantile (QQ) plots are constructed only for
Lahore station in Figures 3, 4 and 5 to see whether the hypothesized distributions are a suitable model
for extreme rainfalls and deviation from the observed data indicates incompatibility of the model. The
Plots indicate that GEV and GP distributions are most reasonable for extreme rainfalls for four
locations as compared to Gamma distribution. Moreover, the density plots of GEV and GP
distributions are also indicated that GEV and GP distributions are appropriate for describing the
behavior of extreme rainfall in Pakistan. These plots and p-values of the three goodness of fit test
statistics are a final confirmation of the selected models.
Table 5:
Goodness of fit tests
GEV Distribution
Stations KS(ML) KS(PWM) AD CS
p-value p-value p-value p-value
Islamabad 0.8155 0.8729 0.8000 0.69519
Murree 0.6276 0.6816 0.6160 0.8431
Lahore 0.9777 0.9986 0.1892 0.9864
Sialkot 0.1193 0.4645 0.9975 0.6915

426 Kamran Abbas, Alamgir, Sajjad Ahmad Khan,
Dost Muhammad Khan, Amjad Ali and Umair khalil
Gamma Distribution
Table 6:
Goodness of fit tests - continued
Stations KS(ML) KS(PWM) AD CS
p-value p-value p-value p-value
Islamabad 0.8214 0.7902 0.5360 0.7760
Murree 0.7074 0.8859 0.4804 0.9338
Lahore 0.1786 0.1150
Sialkot 0.5586 0.5475 0.1862 0.1300
GP Distribution
Stations KS(ML) KS(PWM) AD CS
p-value p-value p-value p-value
Islamabad 0.5120 0.5439 0.9850 0.8713
Murree 0.920 0.8230
Lahore 0.9777 0.9941 0.8650 0.7263
Sialkot 0.5432 0.6572 0.8152 0.7719
Figure 3:
Diagnostics for GEV distribution for Lahore station: (a) Probability Plot (b) Quantile Plot (c) Return
Level Plot (d) Density Plot

Modeling the Distribution of Annual Maximum Rainfall in Pakistan 427
Figure 4:
Diagnostics for GP distribution for Lahore station: (a) Probability Plot (b) Quantile Plot (c) Return
Level Plot (d) Density Plot
Figure 5:
Diagnostics for Gamma distribution for Lahore station: (a) Probability Plot (b) Quantile Plot
4.3. Return Period
Return level estimates and associated 95% confidence interval for different years are calculated by
using the quantile function of GEV distribution, whereas confidence intervals are constructed using the
profile likelihood method and results are given in Table 6 for comparison purpose. Return levels
clearly indicate that among the four stations Murree has the highest return levels for 2, 5, 10, 20 and 30
years return period but only for 50 years return period Sialkot and Islamabad has the highest return

428 Kamran Abbas, Alamgir, Sajjad Ahmad Khan,
Dost Muhammad Khan, Amjad Ali and Umair khalil
levels. The return level estimates based on GEV distribution are close to the observed values of annual
maximum daily rainfall. This condition is also reinforced by return level plot (Figure 3, c) suggesting
that return level estimates are consistent to the observed values. Results also reveal that the significant
differences of return levels among four stations.
Table 7:
Return level estimates using GEV distribution
Return level estimates
[95 confidence interval]
Stations 2 5 10
Islamabad 240.68 368.08 453.27
[203.11, 278.25] [312.81, 423.35] [376.40, 530.13]
Murree 334.76 451.32 515.57
[296.85, 372.68] [406.68, 495.95] [463.95, 567.20]
Lahore 46.92 99.86 165.53
[36.58, 57.67] [66.05, 133.67] [83.98, 247.08]
Sialkot 62.45 153.13 264.51
[40.24, 84.66] [94.58, 211.68] [104.35, 424.68]
Stations 20 30 50
Islamabad 535.61 583.26 643.11
[426.62, 644.59] [450.27, 716.24] [474.36, 811.87]
Murree 569.05 596.61 628.10
[507.08, 631.02] [526.96, 666.26] [547.29, 708.90]
Lahore 269.38 356.70 506.92
[85.4, 453.36] [69.82, 643.58] [18.33, 995.53]
Sialkot 439.24 585.28 835.20
[40.99, 837.49] [68.51, 1229.08] [296.16, 1966.57]
5. Conclusion
The aim of this paper is to compare GEV, Gamma and GP distributions in fitting extreme rainfall
amount. The results of the best fitting distribution may differ for a particular station. Kolmogrove
Simirnov (KS), Chi
-
Squared (CS) and Anderson Darling (AD) goodness of fit tests were used to
identify best fit at 5% level of significance. Based on these findings we concluded that GEV and GP
distributions are most suitable for extreme rainfalls. These results are in agreement with the result
obtained by Zin et al. (2008) and Azumi et al. (2010). According to Zin et al. (2008) GEV and GP
models are well fitted for extreme rainfall in Peninsular Malaysia. Azumi et al. (2010) also found that
GP distribution is also most suitable model for rainfall intensity. Overall, this research also
demonstrates that how applying parametric distributions with parameter estimates using extreme
rainfall historical data. These results can be encouraging to step
-
up the information about the rainfall
history for particular area, local farmers, policy makers, scientists studying precipitation and storm
water management planning.
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