International Journal of Scientific Engineering and Applied Science (IJSEAS) – Volume-3, Issue-11, Nov 2018
ISSN: 2395-3470
www.ijseas.com
A Markov Model for Prediction of Annual Rainfall
*A.O. Ibeje
1
, J. C. Osuagwu
2
and O. R. Onosakponome
2
1
[email protected], Department of Civil Engineering, Imo State University, P.M.B. 2000, Owerri, Nigeria.
2
Department of Civil Engineering, Federal University of Technology, Owerri, Nigeria
Abstract
Markov modelling is one of the tools that can be
utilized to assist planners in assessing the rainfall.
The first-order Markov chain model was used to
predict annual rainfall intervals using transitional
probability matrices and Monte Carlo simulation.
The class intervals were treated as probability states
and the transition probability matrices were using the
frequency distribution of the class intervals. Six
probability states were used to construct first-order
Markov chain. To illustrate the applicability of the
model, Owerri city in Nigeria, was used as a case
study. Model parameters were estimated from 1976-
2005 historical rainfall records. When Monte Carlo
scheme was used to simulate the probability of
occurrence of annual rainfall intervals, the results
demonstrated that the model can be applied
successfully and provided forecasts of high accuracy.
Keywords: Annual rainfall, Markov Chain, Model,
Monte Carlo, transition probability,
1. Introduction
Water plays an important role for agriculture,
industry and daily domestics. It is one of the vital
natural resources. The amount of rainfall received
over an area is an important factor in assessing the
amount of water available to meet the various
demands of agriculture, industry and other human
activities [1]. The analysis of rainfall records for long
periods provides information about rainfall patterns
and variability [2].Prediction of the probability of
occurrence of annual rainfall has remained neglected
despite its great relevance in hydrologic risk and
reliability studies. The variability of rainfall is very
important for agriculture. The estimation of annual
rainfall occurrence from available time series helps to
obtain predictions for annual rainfall statistical
parameters such as the averages, standard deviations
and the coefficient of variation. Several methods
([1];[2];[3];[4]&[5]) have been proposed by various
researchers for modelling rainfall data. Mandal et al.
[6] analyzed the annual rainfall of Daspalla region in
Odisha, eastern India for prediction of monsoon and
post-monsoon rainfall and the results revealed that
Log Pearson Type-III and Gumbel distribution were
the best fit probability distribution functions.
Shadeed et al. [7] evaluated the distribution
characteristics of the rainfall in semi-arid regions and
Gumbel distribution was proved to simulate the
annual rainfalls of the six stations of Faria catchment
in India. Yet, most of these models do not account
for the year to year variations in the probability of
occurrence of rainfall. These variations may either be
in the form too much water, which will lead to
flooding or too little water, which will lead to
draught.
A stochastic model can be used to predict the
probability of occurrence of annual rainfall.
According to Taha [8], a stochastic process
consisting of a random variable X
n, characterizing the
state of the system at discrete points in time t=1,2,.. is
a Markov process if the occurrence of a future state
depends only on the immediately preceding state.
The Markov chain models provide quick forecasts
immediately after any observations have been made
because they use only the local information as
predictors and they need minimal computation after
the data have been processed [8]; [9].The main
purpose of the present study is to show the use of
first-order Markov Chain model for the prediction of
the occurrence of annual rainfall intervals in Owerri,
Nigeria.
2. Materials and Method
2.1 Study Area
As shown in Fig 1, the study area is Owerri. Owerri
is the capital city of Imo State located in South-East
of Nigeria. Owerri situates between 50 20'N, 6055'E
in the south-western corner and 50 34'N, 70 08'E in
the north-eastern corner [10]. It falls within the
rainforest zone of 2290mm of rainfall per annum,
relative humidity of 55-85% and temperature of 27
0
C. 1

International Journal of Scientific Engineering and Applied Science (IJSEAS) – Volume-3, Issue-11, Nov 2018
ISSN: 2395-3470
www.ijseas.com
It has a subequatorial climate. The two prevalent
seasons, the dry and rainy seasons occur from
October to March and April to September
respectively [10]. Owerri covers a land mass of
5200km
2
and lies entirely within coastal plain sand
stones.
Fig. 1 Location Map of Owerri in Imo State, Nigeria
2.2 Data Collection and Processing
1976 to 2005monthly rainfall amount of Owerri were
obtained from the Nigerian Meteorological Agency
(NIMET) office at Imo Airport, Owerri. The daily
data were summed to yield the annual rainfall amount.
2.3 Determination of Annual Rainfall States
A frequency table of annual rainfall is obtained by
dividing the range of data into non-overlapping bins.
According to Taha [8], given the boundaries (I
i-1, Ii)
for bin i, the corresponding frequency is determined
as the count of all the raw data, x, that
satisfy
. Bin i are then converted into
state i such that
2.4 Model Formulation
Given the chronological times the
family of random variables
is said to be a Markov
process if it possesses the following property:
(1)
This is known as the one-step transition probability
of moving from state i at t-1 to state j at t i.e
. By
definition,
(2)
2.5 Param
eter Estimation
The estimates of
was obtained by
(3)
where is the number of times the observed data
went from state i to state j.
Wilks [11] defined the transition probability matrix P
for n-step first-order Markov chain as
2.6 Model Simulation
For
generating the sequences of annual rainfall states,
the initial state, say state i, was selected randomly.
Then random values between 0 and 1 were produced
by using a uniform random number generator [9].
This is called Monte Carlo simulation. For next
annual rainfall state in first-order Markov process,
the value of the random number was compared with
the elements of the i
th
row of the cumulative
probability transition matrix. According to Piantadosi
et al. [9], if the random number value was greater
than the cumulative probability of the previous state
but less than or equal to the cumulative probability of
the following state, the following state was adopted.
The cumulative probability transition matrix was
obtained by successive multiplication of P matrix by
itself, until a stabilization of the transition
probabilities led to the transition probability matrix. 2

International Journal of Scientific Engineering and Applied Science (IJSEAS) – Volume-3, Issue-11, Nov 2018
ISSN: 2395-3470
www.ijseas.com
Thus if the transition probability in the i
th
row at the
k
th
state was Pik, then the cumulative probability, Pik,
could be expressed as (5)
3. Results and Discussion
Fig. 1 shows the variation of annual rainfall in
Owerri between 1976 and 2005. The total annual
rainfall ranged between 2889.9 to 527.6 mm with a
mean of 2319.7 mm and the standard deviation is
320.765 (Table 1).The Kurtosis and coefficient of
skewness indicate platykurtic (kurtosis < 3) and
negative skewness, which means that the probability
plot has a flat tail at the left side. The data has 15%
variability indicated by the coefficient of variation.
Fig. 2 Annual Variation of Owerri Rainfall from 1976 to
2005

A, B, C, D, E and F are the rainfall states obtained from the frequency distribution table shown in Table 2.
Table 1: Descriptive Statistics of Annual Rainfall
Data of Owerri
Table 2: Frequency Distribution of Annual Rainfall
Data
Probability transition
matrix P defines the Markov
Chain presenting the six states of annual rainfall was obtained from equation (4) as
The cumulative probability transition matrix, P
c
obtained from equation (3) as the cumulative
summation of each row as
The transition probabilities show that if this year’s
rainfall interval is 0-1527.6mm(state A), there is a
40% chance that it will not vary next year, 40%
chance that it will vary between
1527.6-1800.06mm
(state B), and 40% chance that it will range
between2344.98-2617.44 (state E).If this year’s
annual rainfall interval is 1800.06-2072.52 mm (state
B), there is 18% chance that it will be in state A next
year, 18% chance that it will not vary, 36% chance
that it will be in state A, and 9% chance that it will be
in states D, E or F. If this year’s annual rainfall
interval 1800.06-2072.52 mm (state C), there is 22%
chance that it will vary between 0-1527.6 mm (state
A)next
year, 44% chance that it will vary in range
of1800.06-2072.52 mm (state B), 22% chance that it
will not vary, and 11% chance that it will range
between 2072.52-2344.98mm (state D).
Class Frequency State
0-1527.6 1 A
1527.6-1800.06 2 B
1800.06-2072.52 2 C
2072.52-2344.98 9 D
2344.98-2617.44 11 E
>2617.44 5 F
Statistic Mean
Standard
Deviation Skewness Range
Sample
Size
Value 2319.7 320.765 -0.7283 1362.3 30 3

International Journal of Scientific Engineering and Applied Science (IJSEAS) – Volume-3, Issue-11, Nov 2018
ISSN: 2395-3470
www.ijseas.com

In Table 3, by Monte Carlo simulation, the change of
annual rainfall intervals where generated as the
random number changes values and assumes the
various states of rainfall data. The synthetic annual
rainfall series were obtained from the states by
decomposing the states; where u is a random number
from a uniform distribution u



A if u < 0.1818
B if 0.1818 < u < 0.3636
C if 0.3636 < u < 0.7272
State = D if 0.7272 < u < 0.8181
E if 0.8181 < u < 0.909
F if 0.909 < u < 1.0





4. Conclusions
The results of the study show that Markov chain is
useful in simulating future rainfall intervals. This
type of simulation is useful in generating intervals of
rainfall when the raingauge malfunctioned during
small interval of time. The analysis of yearly rainfall
shows that Markov Chain approach provides one
alternative of modelling future variation in rainfall.

















































References
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Table 3: Synthetic Generation
of Annual Rainfall
Year
Random
Number State
1976 2401.7 B
1977 1988.8 D
1978 2336.7 C
1979 2685.9 A
1980 2440 B 4

International Journal of Scientific Engineering and Applied Science (IJSEAS) – Volume-3, Issue-11, Nov 2018
ISSN: 2395-3470
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