Fitting daily rainfall amount in Peninsular Malaysia using several types of exponential distributions

Abstract

This paper presents a statistical study on fitting distribution of rainfall amount in Peninsular Malaysia using several types of exponential distributions for periods from 21 to 35 years. The mixed gamma, mixed weibull and mixed exponential are proposed and tested together with their single distributions to identify the optimal model for daily rainfall amount in several rain gauge stations in Peninsular Malaysia. The selected model will be determined based on the minimum error produced by some criteria of goodness of-fit (GOF) tests. The results indicated that the mixture distributions are better than the single distributions in modeling rainfall amount where the mixed weibull is identified as the most appropriate model for the majority of sites in Peninsular Malaysia. These results however are varied between the rain gauge stations which are strongly influenced by their geographical, topographical and climatic changes

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Journal of Applied Sciences Research, 3(10): 1027-1036, 2007
© 2007, INSInet Publication
Corresponding Author:Jamaludin Suhaila, Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia,
81310, Skudai, Johor, Malaysia.
E-mail: suhaila@mel.fs.utm.my Tel: (607)-553 4317, Fax: (607)-556 6162
1027
Fitting Daily Rainfall Amount in Peninsular Malaysia Using Several
Types of Exponential Distributions
Jamaludin Suhaila and Abdul Aziz Jemain
12
Department of Mathematics, Faculty of Science,
1
Universiti Teknologi Malaysia, 81310, Skudai, Johor, Malaysia.
School of Mathematical Sciences, Faculty of Science and Technology,
2
Universiti Kebangsaan Malaysia, 43600, Bangi, Selangor, Malaysia.
Abstract: This paper presents a statistical study on fitting distribution of rainfall amount in Peninsular
Malaysia using several types of exponential distributions for periods from 21 to 35 years. The mixed
gamma, mixed weibull and mixed exponential are proposed and tested together with their single
distributions to identify the optimal model for daily rainfall amount in several rain gauge stations in
Peninsular Malaysia. The selected model will be determined based on the minimum error produced by
some criteria of goodness of-fit (GOF) tests. The results indicated that the mixture distributions are better
than the single distributions in modeling rainfall amount where the mixed weibull is identified as the most
appropriate model for the majority of sites in Peninsular Malaysia. These results however are varied
between the rain gauge stations which are strongly influenced by their geographical, topographical and
climatic changes.
Keywords: Mixture of two distributions; goodness-of-fit test; mixed exponential, mixed weibull;
mixed gamma.
INTRODUCTION
Modeling daily rainfall data using various
mathematical models has been an important research in
hydrology for the last 30 years. The use of
mathematical models of rainfall has been applied
worldwide in order to give a better understanding about
the rainfall pattern and its characteristics. This process
involves the study on the sequence of dry and wet
days as well as the study of rainfall amount on the wet
days. Markov chain models are normally used to model
the sequence of dry and wet days while rainfall
[1,2,3,4]
amount is often modeled using a two-parameter gamma
distribution. Other theoretical distributions that
[5,6,7,8,9]
have also been employed in the analysis of
rainfall are exponential distribution, kappa
[10]
B
distribution, S distribution, mixed exponential
[11][12]
distribution , weibull distribution and skew
[13,14,15,16][17]
normal distribution .
[14,15]
The selection of the best fitting distribution has
always been a key interest in the study of rainfall
amount. Suhaila and Jemain compared several types
[18]
of normal transform distributions to fit daily rainfall
amount in Peninsular Malaysia. They found that the
mixture of two lognormal distributions gave the best
fitting model. In this paper, we will focus on mixture
distributions which come from several types of
exponential distributions that will be employed together
with their single distributions in order to find the best
model in fitting rainfall data. In order to verify the
suitable distribution that best describes the rainfall
amount, the new method of goodness-of-fit tests (GOF)
based on the likelihood ratio statistics which has been
developed by Zhang and Zhang and Wu will be
[19][20]
engaged together with the traditional GOF tests. An
additional criterion that is included in the analysis is
the median of absolute difference between the
hypothesized and empirical distribution function. The
final result on the best fitting distribution will be
chosen based on the minimum error specified by these
GOF criteria.
MATERIALS AND METHODS
Case Study: Daily rainfall series data for this study
have been obtained from the Malaysian Meteorological
Department for the periods ranging from 21 to 35
years. For this study, eighteen rain gauge stations were
chosen based on the completeness of the data. The
details are shown in Table 1.The stations are
selected to represent rainfall pattern for the
whole Peninsular Malaysia.

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Table 1:The latitude, longitude and period of records data obtained for each of the eighteen rain gauge stations.
CodeStationsAltLatitudeLongitudePeriod of records
1 Senai37.8 m1°38 N103°40 E1974-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2Kluang88.1 m2°01 N 103°19 E1974-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3Malacca8.5 m2°16 N102°15 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
4Mersing43.6 m 2°27 N103°50 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
5Petaling Jaya60.8 m3°06 N101°39 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
6Subang16.5 m3°07 N101°33 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
7Temerloh39.1 m3°28 N102°23 E1982-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
8Kuantan15.3 m3°47 N103°13 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
9Batu Embun59.5 m3°58 N102°21 E1978-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
10Sitiawan7.0 m4°13 N100°42 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
11Cameron Highlands1545 m4°28 N101°22 E1983-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
12Ipoh40.1 m4°34 N101°06 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
13Bayan Lepas2.8 m5°18 N100°16 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
14Kuala Trengganu5.2 m5°23 N103°06 E1985-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
15Kuala Krai68.3 m5°32 N102°12 E1984-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16Kota Bharu4.6 m6°10 N102°17 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
17Alor Star3.9 m6°12 N100°24 E1971-2005
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
18Chuping21.7 m6°29 N100°16 E1979-2005
There are major differences of climate observed
within the country especially between the west and east
coasts of Peninsular Malaysia and slightly less so
between the north and south. These differences arise
from the discrepancy of altitude and the exposure of
the coastal lowlands to the southwest and northeast
monsoon winds. The southwest monsoon usually occurs
in mid of May and ends in September. The wind is
generally light going below 15 knots. On the
other hand, the northeast monsoon usually begins
in early November and ends in March with a
speed between 10 to 20 knots. During this season,
the more severely affected areas are the east coast
states of Peninsular Malaysia where the wind
may reach more than 30 knots. The coasts that are
exposed to the northeast monsoon in Peninsular
Malaysia tend to be wetter than those exposed to the
southwest monsoon. The period of the southwest
monsoon is a drier period for the whole country,
particularly for the other states of the west coast of the
Peninsula. The period of change between the two
monsoons is a transitional period which occurs in April
and October. These two transitional periods often result
in heavy rainfall which usually occurs in the form of
convective rains. During these periods, the west coast
is generally wetter than the east coast. Thus, in general
the rainfall distribution in Peninsular Malaysia is
mostly affected by those two monsoons and the two
transitional periods.
Modeling Daily Rainfall Amount: Five models of
rainfall amount are described as follows with their
probability density functions. Note that X is the random
variable representing the daily rainfall amount.
The gamma distribution with two parameters, á
and â denote the shape and scale parameters
respectively.
The shape parameter governs the shape of the
rainfall distribution and the scale parameter determines
the variation of rainfall amount series which is given
in the same unit as the random variable X.
The weibull distribution with two parameters, á
and â denote the shape and scale parameters
respectively.

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J. Appl. Sci. Res., 3(10): 1027-1036, 2007
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Slightly similar to the gamma distribution, the
weibull distribution shares the ame properties of
nonnegativity and positive skewness. The weibull and
gamma distribution are identical for á = 1 , where both
of these distributions reduce to the exponential
distribution.
The mixed exponential distribution with three
parameters is a mixture of two one-parameter
exponential distributions where p denotes the mixing
probability which determines the weights given to the
1
two exponential distributions with scale parameters â
2
and â .
The mixed exponential distribution has the same
characteristic as the single parameter exponential where
the scale parameters for both components represent
the variation of rainfall amount series which have the
same values as random variable X. Large values of
scale parameters give large variation of rainfall
amount series.
The mixed weibull with five parameters is a
mixture of two two-parameter weibull distributions
where p denotes the mixing probability which
determines the weights given to the two weibull
12
distributions. Here parameters á and á represent the
12
shape parameters whereas â and â represent the scale
parameters in the distribution.
The mixture of two weibull distribution can be
reduced to a single parameter of exponential
12
distribution if we set p = 1 with either á = 1 or á =1.
The mixture of two weibull distributions share the
same properties as the single weibull distribution where
their shape parameters determine the shape of the
rainfall distribution for both components while the
scale parameters again represent the variation of
rainfall amount series which has the same units as
the random variable X. The differences between the
first and second components are determined by their
mixing probability.
The mixed gamma with five parameters is a
mixture of two two-parameter gamma distributions
where p denotes the mixing probability which
determines the weights given to the two gamma
distributions where á ,á ,â and â denote the shape and
1 21 2
scale parameters respectively.
Once again, the mixture of two gamma
distributions shares the same properties as the single
gamma distribution. The shape and scale parameters
represent the same properties as we discussed before.
So we did not elaborate further.
Estimation Procedure: Many methods are available
for parameter estimations, which include the method of
moments (MM), maximum likelihood estimation
(MLE), the least squares method (LS), L-moments and
generalized probability weighted moments (GPWM).
The MLE method is considered in this study because
it provides the smallest variance as compared to other
methods. The idea of this method is to find a set of
parameters that will maximize the likelihood function.
The parameters are obtained by differentiating the log
likelihood function with respect to the parameters of
the distribution.
Gamma: The logarithm of the likelihood function of
the considered distribution is given as follow.
The MLE for the gamma distribution is easily
calculated using the two approximation methods as
described by Thom and Greenwood and Duran.
[21][22]
However, in this paper the maximum likelihood
equations for this distribution were solved by using the
quasi-Newton algorithm in the nonlinear programming.
The formula shown in Eq. (10) from the MM method
was used to determine the initial values of the

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1030
parameters for the iterative procedure in the calculation
of the gamma density function.
The quasi-Newton algorithm once again was
applied to solve the nonlinear equation given by Eq.
[
(11). The method of moments given by Wong was
23]
used to determine the initial values for the iteration.
The MLE estimations of the single gamma and
weibull distributions are easier to compute since
many relevant softwares are available to compute
the gamma and weibull MLE directly such as
SPLUS and MATLAB.
Mixture of two exponential distribution
The maximum likelihood equation for this
distribution is in implicit form. Once again the quasi-
Newton algorithm has been used to solve Eq. (12) with
the restrictions as described in Eq. (4). The iterative
procedure requires initial values where they were
estimated by the method of moments as shown by
Rider and some other initial values as suggested
[24]
below.
Mixture of two weibull distribution
In this paper, the MLE method has been used to
solve the log likelihood equation as expressed in Eq.
(14). We used the quasi-Newton algorithm to resolve
this nonlinear equation, with the restrictions as
described in Eq. (6).
We next consider the initial values for the iterative
procedure. Regarding to the restrictions in Eq. (6), we
tried several values given in Eq. (15) by trial and error.
á1 ? á2, â2 ? â1 and p = 0.2,0.6,0.8 (15)
Another alternative is to use the initial values as
shown in Eq. (16) that was given in the study done by
1
Carta and Ramirez where m and s refer to the
[25]2
mean and variance of rainfall data.
Using both approaches, we found that they give
the same results for the estimated parameters.
Mixture of two gamma distribution
Again the MLE method was used to solve the
following nonlinear equation;
The same estimation procedures for the mixture of
two gamma distribution were carried out to determine
their estimate parameters following the restriction
specified in Eq. (8) by trying some initial values as
suggested in Eq. (15).
Goodness-of-fit Tests (GOF): Seven different GOF
tests have been used in this study to identify the best
fit models. The tests are based on the degree of
similarity between the empirical distribution function
(1)
Fn(x ) and the hypothesized distribution function
(1)
Fn(x è). The chosen distribution that best fits the
daily rainfall amount is based on the minimum error
indicate by all these seven tests.
The first criterion involves the median absolute
difference (MAD) between The
formula is given as below.
In this equation, ?represents the ordered data
whereas the vector of estimated parameters is
represented by .
The second criterion involves the classical EDF
statistics which are also called as classical GOF tests.
They are usually divided into two classes; the
supremum and the quadratic. This paper concentrates
on the well known Kolmogorov-Smirnov statistic D
which belongs to the supremum class of EDF statistics.

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This statistic is based on the largest vertical differences
between . For the quadratic
class, this paper focuses on the Cramer-von-Mises
statistic W and the Anderson Darling statistic A test.
22
This class of statistics is based on the squared
difference .
(i)
Let with x represents the
ordered data whereas the vector of estimated
parameters is represented by . Then the above EDF
statistics have the following formula:
The third criterion involves new powerful
GOF tests which are based on the likelihood
ratio statistics between the hypothesized distribution
and the empirical distribution
(i)
function Fn (x ). The computed formulas used in this
paper are shown in Eq. (22)-Eq. (24) as in Zhang[19]
and Zhang and Wu.
[20]
a) New Kolmogorov-Smirnov test
b) New Cramer-von-Mises test
c) New Anderson-Darling test
RESULTS AND DISCUSSIONS
First, we will have a brief discussion on the
descriptive statistics for each of the eighteen rain gauge
stations and then proceed to give comments on the
results of fitting distributions that are based on GOF
criteria. Finally the remarks on the estimated
parameters for the best model will be made.
Descriptive Statistics: A summary of the basic
statistics of daily rainfall amount for each of the
eighteen rain gauge stations is summarized in Table 2
where the mean, standard deviation, coefficient of
variations, skewness, kurtosis, number of wet days and
maximum amount of daily rainfall of each station are
given. Kota Bharu station received the highest mean
daily rainfall amount followed by three other stations
which are Kuantan, Kuala Trengganu and Mersing. In
terms of the standard deviation, these four stations once
again showed high values which indicate a large
variation in their daily rainfall amount series. We
noticed that all of these four stations are located along
the east coast of Peninsular Malaysia. As we mentioned
before, the entire east coast is exposed to the northeast
monsoon which is known to bring heavy rainfall during
the monsoon. According to Zalina, normally 45 to 55
[26]
percent of the annual maximum rainfall events for
those stations were experienced during this period and
the rainy events during this period are very long with
heavy and moderate rains occurring intermittently.
Therefore, those areas are subjected to flooding during
the monsoon. Meanwhile, Sitiawan station indicates the
lowest rainfall amount. This probably occurred because
of the northeasterly winds are blocked by the Main
Range (Banjaran Titiwangsa) that affect most of the
stations along the west coast of the Peninsular. That
explains why the west coast is drier than the east coast
in terms of rainfall amount.
The irregularity of the daily rainfall between
stations is represented by the coefficient of variation
(CV) which is evident in all cases that the 100% is
clearly exceeded. Stations which are located in the east
coast have shown high variability of rainfall amount
which ranged between 180% and 200% compared to
other stations that ranged between 130% and 160%.
The lowest variability is indicated by Cameron
Highlands station which is located on the Main Range.
Based on the results given in Table 2, this station
indicates the smallest value of skewness and the lowest
maximum amount of daily rainfall. This shows that the
rainfall distribution at this station is more evenly
distributed than other stations.
The amount of rainfall is also found to be
uncorrelated to the number of wet days of the stations.
For example, Kota Bharu station has a smaller number