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Volume-3, Issue-1, January 2011, ISSN No.1998 - 7889 Eastern University Journal
Stochastic Modelling of Daily Rainfall at Naogaon District in
Bangladesh: A Comparative Study
M. A. B. Barkotulla*
M. Sayedur Rahman**
Abstract
The weather plays a significant role in the agriculture of a country. Rainfall is the most
important weather parameter affecting non-irrigated crop areas. The region lies in the
driest part of the country in terms of rainfall and is semi-arid in character. Markov
chain model has been used to evaluate probabilities of getting a sequence of wet-dry
weeks over this region.  A non-linear proceed, the model is tested using multivariate
logistic regression technique. The results showed the relative frequency of rain against
the estimated value of P using the logistic regression at 7 days interval. The results of
the relative frequencies of 0.29 and 0.71 are similar in the study area. The logistic
regression technique allows flexibility in deciding between areas of indicated rain or no
rain. The results indicate that no rain is predicted when the estimated value of P is less
than 0.5 and rain, when it is greater than 0.5. Therefore when the estimated P is less
than 0.1, it is almost not raining. Basically this type of information gives the advantages
of logistic regression over a simple threshold technique to agricultural planners and
irrigation engineers in identifying the areas where agricultural development should be
focused as a long-term drought mitigation strategy.
Key word: Rainfall, Markov chain, logistic regression, probability, parameter,
relative frequency.
1. Introduction
Agro-ecologically the Barind Tract is divided into three regions: Level Barind Tract,
High Barind Tract and Northeastern Barind Tract. The High Barind Tract is located in
Rajshahi, Nawabganj and Naogaon districts. The region lies in the driest part of the
country and is semi-arid in character. The mean annual rainfall is about 1,350 mm
(NAP, 2005). Groundwater supplies are generally poor in the more hilly western part.
The predominant land use is transplanted aman grown as a single crop during the rainy
season. The rest of the year is arid and basically crop less. Non-irrigated land generally
remains fallow in the dry season. Early rabi crops are grown where irrigation is
available. Shortage of rainfall affects the critical reproductive stages of transplanted
aman crops in October, reducing its yield, particularly in those areas with low soil
moisture holding capacity (WARPO- EGIS, 1996).
* Associate Professor, Department of Crop Science and Technology, University of Rajshahi
** Professor, Faculty of Business Administration, Eastern University
Stochastic Modelling of Daily Rainfall at Naogaon District in Bangladesh: A Comparative Study
2
Weather is a natural variability but seems to strive after effects of the global warming.
Severe weather such as heavy rains causing flash flood is more frequent or at the least
more frequently reported, while a prolonged sequence of dry cyclones (those passing
without rains but with clouds) causing drought is more common than ever before (Kim,
2000).
Changes in global temperatures and precipitation have enormous impacts, such as on the
geographical range of crop cultivation; this type of information is essential when
discussing the impacts of global warming on humanity. Variability of rainfall is
especially important because it effects on both hydrology and agriculture of a drought
prone high Barind region in Bangladesh (Rahman and Alam, 1997, Barkotulla et al.,
2007)
Rahman (1999b) developed a simulation rainfall model and tested it using a multivariate
logistic regression technique to estimate the probability of rain events.  The model used
historical data to estimate model parameters.  The estimated parameters could be used in
the model to produce a daily-simulated rainfall.  The Markov chain model is useful to
have a measure of the certainty of the decision.  This information may be useful for both
instantaneous and climate time scale.
Several studies have demonstrated significant changes in local, regional, and global
surface air temperature and precipitation patterns measured over the past century
(Vinnikov et al., 1987; Jones, 1988; Hensen and Lebedeff, 1988; IPCC, 1997).  Climatic
system is not simple to be understood due to its non-linearity of interactions among
many physical and dynamical processes under external forcing (Oh and Lee, 2000).
After 1971 Bangladesh has experienced droughts of major magnitude in 1973, 1978,
1979, 1981, 1982, 1989, 1992, 1994, and 1995. Although droughts are not always
continuous in any area, they do occur sometimes in the low rainfall zones of the country.
As listed above, Bangladesh experienced consecutive droughts in 1978-1979, 1981-
1982, and 1994-1995. The droughts of 1994-95 in the northwestern districts of
Bangladesh led to a shortfall of rice production of 3.5 million tons (Paul, 1995). Two
critical dry periods are distinguished (Karim et al., 1990). Rabi and pre-Kharif drought
(January - May), due to: (i) the cumulative effect of dry days; (ii) higher temperatures
during pre-Kharif (> 40 degrees Celsius in March-May); and (iii) low soil moisture
availability (NAP, 2005).
2. Description of the Study Area
Naogaon district is the western part of Bangladesh with an area of 3435.67 sq km, which
is bounded by West Bengal of India on the north, Natore and Rajshahi districts on the
south, Joypurhat and Bogra districts on the east, Chapai Nawabganj district and West
Bengal (India) on the west. Most of the people of the area are related to agriculture and
agricultural labour. There are three seasons in a year: summer, monsoon and winter. The
summer covers the period from March to May. The monsoon season is spread over the
months of June to October. Eighty percent of the rainfall occurs during the monsoon
M. A. B. Barkotulla, M. Sayedur Rahman
3
season. The winter season is spread over the months of November to February of the
following year. There are hardly any rains during the winter season.
The rainfall is either monsoonal, inter-monsoonal or cyclonic in origin
(http://www.banglapedia.net/HT/C_0288.HTM). Rainfall varies from place to place as
well as year to year. The minimum annual rainfall is 1002.4 mm at Mohadevpur in the
year 2001 and the maximum rainfall is 2421.9 mm at Badolgachi in the year 1998 and
the annual average rainfall 1607.68 during the study period. The annual temperature is
maximum 37.8°C although occasionally it rises up to 400C and the minimum is 11.2°C
sometimes it falls to about 5°C. It is the driest part of the country which has been already
identified as the high Barind region (agro-ecological zone no. 26).
Source: www.banglapedia.net
Fig.1: Map of the study area
Stochastic Modelling of Daily Rainfall at Naogaon District in Bangladesh: A Comparative Study
4
55%
20% 25% Single crop 25%
Double crop 55%
Treble crop 20%
Fig.2: The percentage of agricultural land use for crop production
2.1  Data Collection
There are six rainfall stations located at Naogaon district of high Barind region in
Bangladesh. The stations are Naogaon, Sapahar, Mohadevpur, Badolgachi, Najipur
(Patnitala) and Nitpur (Porsha) (Fig. 1).  The daily rainfall data for 8 years during the
period 1994-2001 of the six rainfall stations in Naogaon district have been considered in
this paper. The secondary data of daily rainfall (mm) were collected from Bangladesh
Water Development Board. In Bangladesh, there are three agro-climatic seasons, namely
(1) Pre-kharif (summer) which includes the pre-monsoon months of March to May, (2)
Kharif which includes monsoon months of June to October in which 80% of the total
rainfall occurs (Rahman and Alam, 1997), and (3) Rabi (winter) season which starts
from November to February.
2.2  Multivariate Logistic Regressions
The significance of rainfall analysis has been highlighted in hydrological and
climatological studies because of its influence in all human activities such as
agricultural, industrial and domestic. Using multivariate logistic regression technique to
estimate the probability of rain event, the probability of rain can be used to map regions
of very likely rain; that probability is close to 1, possibly rain and very unlikely rain, the
probability is close to 0.
M. A. B. Barkotulla, M. Sayedur Rahman
5
2.2.1  Procedures
Multivariate logistic regression is used to predict dichotomous events.  Let P(X1, X2, …,
Xk) be the probability that it is raining.  Let the logistic response function P is the form
P(X1, X2, .…, Xk) = exp(f)/{1+exp(f)},
Where,  f = 0 +jXj.
It is a monotonic function of f and its value lies between 0 and 1.  Dichotomous
variables Y may be defined as
Y =


raining
rainingnot
,1
,0 (1)
Then consider a set of vectors, Xi/ = ( Yi, X1i, X2i,…, Xki),  i=1,2,…,n.  To
estimate the  ‘s, the following procedure can be used,
P(Yi =1) = Pi = exp(f)/{1+exp(f)} (2)
P(Yi = 0) =1-Pi
The joint distribution of Y1,Y2, ……,Yn is given by
g(y1, y2,……,yn) =  Piyi (1-Pi)1-yi (3)
To minimize the negative log of g is equal to maximize the joint distribution function.
Therefore, the loss function is
loss = - [Yi log(Pi) + (1-Yi) log(1-Pi)] (4)
A two state Markov Chain method involve the calculation of two conditional
probabilities: (1) , the probability of wet day following a dry day, and (2) , the
probability of dry day following a wet day. The wet-dry probabilities are not available; it
can be estimated using the average number of days of rainfall,
Pj (W/D) =  . dpj / dmj (5)
Pj (W/W) = 1 -  + Pj (W/D) (6)
where, Pj (W/D)   is the probability of rainfall of wet day after dry day in month j, Pj(W/W) is the probability of rainfall of wet day after wet day in month j, dp is the days of
rainfall in month j and  is equal to .75 for the model.  Given the initial wet-dry state, the
model determines stochastically if rainfall occurs.  A random number (0-1) is generated
and compared with the appropriate wet-dry probabilities.  If the random number is less
than or equal to the wet-dry probability, rainfall occurs on that day.  If random number is
greater than the wet-dry probability, no rainfall occurs (Rahman, 1999b).
Stochastic Modelling of Daily Rainfall at Naogaon District in Bangladesh: A Comparative Study
6
3. Results and Discussions
A non-linear proceed, the model is tested using multivariate logistic regression
technique. The estimated probability of P is denoted by    ffP exp1exp 
This is especially interesting since the logistic function is approximately linear in f for P
values between 0.2 and 0.8.  The result is in agreement with the findings of Neter et al.
(1989).  The estimated values of P were then placed in the class interval.  The simulation
results gave estimated values of P against the no-rain / rain values in frequency and the
relative frequency of rain (Rahman, 1999b).
The logistic regression technique allows flexibility in deciding between areas of
indicated rain or no rain. Neter et al. (1989) suggest a decision rule that no rain is
predicted when the estimated value of P is less than 0.5 and rain, when it is greater than
0.5. Rahman (1999b) used daily simulation rainfall data and suggests that if estimated P
is less than 0.1, it is almost not raining. Basically this type of information gives the
advantages of logistic regression over a simple threshold technique. The data used to
predict P and subsequently the results obtained are given in Table 1 to 6. The results
showed the relative frequency of rain against the estimated value of P using the logistic
regression at 7 days interval. The results of the relative frequencies of 0.29 and 0.71 are
similar in the study area.
Table 1. Relationship between the estimated P values using the rainfall data at 7 days
interval and the measured incidence of rain of Naogaon station at Barind region
Naogaon
Estimated values of  P No rain Rain Rain  /total
0.0-0.2 1606 67 0.04
0.2-0.4 310 124 0.29
0.4-0.6 256 241 0.48
0.6-0.8 52 130 0.71
0.8-1.0 15 118 0.89
Total 2239 680
Table 2. Relationship between the estimated P values using the rainfall data at 7 days
interval and the measured incidence of rain of Sapahar station at Barind region
Sapahar
Estimated values of P No rain Rain Rain  /total
0.0-0.2 1889 71 0.04
0.2-0.4 210 84 0.29
0.4-0.6 201 184 0.48
0.6-0.8 50 125 0.71
0.8-1.0 11 94 0.90
Total 2361 558
M. A. B. Barkotulla, M. Sayedur Rahman
7
Table 3. Relationship between the estimated P values using the rainfall data at 7 days interval
and the measured incidence of rain of Mohadevpur station at Barind region
Mohadevpur
Estimated values of  P No rain Rain Rain  /total
0.0-0.2 1635 66 0.04
0.2-0.4 270 108 0.29
0.4-0.6 227 214 0.49
0.6-0.8 64 160 0.71
0.8-1.0 18 157 0.90
Total 2214 705
Table 4. Relationship between the estimated P values using the rainfall data at 7 days
interval and the measured incidence of rain of Badolgachi station at Barind region
Badolgachi
Estimated values of  P No rain Rain Rain  /total
0.0-0.2 1645 56 0.03
0.2-0.4 290 116 0.29
0.4-0.6 271 275 0.50
0.6-0.8 34 85 0.71
0.8-1.0 15 132 0.90
Total 2255 664
Table 5. Relationship between the estimated P values using the rainfall data at 7 days
interval and the measured incidence of rain of Najipur station at Barind region
Najipur
Estimated values of  P No rain Rain Rain  /total
0.0-0.2 1636 58 0.03
0.2-0.4 275 110 0.29
0.4-0.6 259 245 0.49
0.6-0.8 38 95 0.71
0.8-1.0 21 182 0.90
Total 2229 690
Table 6. Relationship between the estimated P values using the rainfall data at 7 days
interval and the measured incidence of rain of Nitpur station at Barind region
Nitpur
Estimated values of P No rain Rain Rain /total
0.0-0.2 1896 64 0.03
0.2-0.4 240 96 0.29
0.4-0.6 223 204 0.48
0.6-0.8 28 70 0.71
0.8-1.0 8 90 0.92
Total 2395 524
Stochastic Modelling of Daily Rainfall at Naogaon District in Bangladesh: A Comparative Study
8
4. Conclusions
Climate change occurs due to natural and anthropogenic disturbances in our
environment. The lack of rainfall and high air temperature are the main causes
hampering agricultural production in this region. The findings of this paper have
provided confidence in the use of modeling techniques for land use in the Barind region.
Application of the results is necessary for the development of an operational
methodology that could be used in land use planning. The emphasis needs to be placed
on site and season-specific technologies and extension messages to farmers, agricultural
scientists, policy makers and the general public.
The logistic regression technique should be considered, where the model is satisfied in
certain conditions and is useful to have a measure of the certainty of the decision. This
information may be useful for both instantaneous and climatic time scale. The data used
to predict P and the results were obtained. The results of the relative frequencies of 0.29
and 0.71 of rain against the estimated value of P using the logistic regression of 7 days
interval are similar in the study area. The probability of occurrence of rainfall is of vital
importance in efficient planning and execution of water programs for agricultural
development and environmental strategies in Bangladesh.
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Stochastic modelling of daily rainfall at Naogaon district in Bangladesh: a comparative study

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