Analysis of the Rainfall Occurrence in Indonesia Region
Using Point Process Model
Nurtiti Sunusia*, Herdiani E.Ta, and Nurdina
a
Department of Mathematics Faculty of Mathematics and Natural Sciences
Hasanuddin University,
Jl. Perintis Kemerdekaan Km 10 Tamalanrea,
Makassar, South Sulawesi, Indonesia
*Corresponding Author: ntitisanusi@gmail.com
ABSTRACT
The aims of this research is study a point process model for rainfall occurrence. The
model is a renewal process. In this model, the rainfall occurrence is considered by the
random points in a space, where every point denote time or/and location of an event.
Conditional Intensity of the model is associated by hazard rate. Parameter of the
model is estimated by hazard rate likelihood point process (HRLPP). The estimation
is applied to a data set of daily rainfall occurrences in two stations in Indonesia.
Keywords: Point Process, Renewal Process, Hazard rate, Conditional Intensity.
1. Introduction
Rainfall is one of the most important inputs for hydrological modeling, and indicators
for climate change impact studies [4]. Rain is the decline events drops of water from
the sky to the earth's surface. Rain is also a water cycle on Earth. The object of this
paper is to describe the probabilistic models for the behavior in time of the
precipitation at a fixed point in space.
There is an extensive literature on stochastic models for rainfall. For some
general comments on the connection with point processes [2]. Stern & Coe (I984)
have a model for daily rainfall in which wet and dry days occur in a Markov chain
with seasonally dependent transition probabilities and in which the amounts of rain
per wet day have a gamma distribution with seasonally dependent parameter.
Some previous researchers have modeled the occurrence of the phenomenon of rain
through several models, among others, Smith in 1985 [1], Smith in 1983 [3], and
Rodriguez in 1987 [2]. In their study, they used the Newman-Scott cluster processes.
In the present paper, we study one model in point process that the conditional
intensity of the process is not depend on history but it depends only on the elapsed
time since the last occurrence. The model is called renewal process and the
conditional intensity coincides with so-called hazard rate. The daily rainfall of data
from two stations are used to simulate the hazard rate.
2. Definition and Notation
Temporal dynamics on a time interval
is represented by an average rate
called intensity. This relates to the expectations of the number of occurrences within a
certain time period or interval. Generally, the likelihood function is the product of
density functions opportunities as proposed by Vere Jones in 1995 [8]. In point
process, the density function is unknown. Therefore, the likelihood function of point
process is approximated by a Poisson principle.
Let
denote the number of event in interval
with
.
The stationary Poisson process in a line is defined by [9]:
(1)
where
or
. So we have,
Based on (1), let there are n events in interval
that are
probability that one event in
and no more event left in
.
so that, for interval
(2)
, then
is
(3)
we have
(4)
Thus, the likelihood of point process is:
(5)
Hazard Rate Likelihood Point Process (HRLPP) have been constructed in Sunusi
(2010) . It can be used to estimate the hazard rate temporal point process.
3. Main Result
3.1 Estimating Conditional Intensity
Let N be a renewal process with inter arrival time distribution F and interarrival time
density f, then the stochastic conditional intensity function is called hazard rate
given by
(6)
where
, is the backward recurrence time at t or the elapsed time at t since
the most recent event.
A Poisson process is a renewal process with exponentially distributed inter
arrival times. Based on equation (6), we have
(7)
where
is the parameter of the exponential distribution and
is the history of
arrivals prior to t. This is consistent with that proposed by Sunusi in 2014 [7].
Because of the stochastic intensity of Poisson processes is constant, so the rate of
occurrence at given time t is not affected by the history of arrivals prior to t. Based on
(5), we have :
.
(8)
where n is the number of occurrence in the interval and
is interval observation.
3.2 Case Study
In this session, a case study for determine the rainfall occurrence is studied. In this
study, we estimated hazard rate of occurrence of rainfall using data which taken from
two meteorological stations, that are meteorological stations of Curug/Budiarto and
Tanjung Priok Jakarta.
Table 1. The Summary of Hazard Rate Value for Station 96739 (Curug/Budiarto,
Indonesia).
No
Interval
n
Hazard Rate (Curug Station)
1
Jan-Mar 09
2
2
0.0333
2
Apr-Jun 09
1
1
0.0172
3
Jul-Sep 09
0
0
0.0000
4
Okt-Des 09
1
1
0.0175
5
Jan-Mar 10
3
3
0.0536
6
Apr-Jun 10
1
1
0.0189
7
Jul-Sept10
1
1
0.0192
8
Okt-Des 10
2
2
0.0392
9
Jan-Mar 11
2
2
0.0408
10
Apr-Jun 11
2
2
0.0426
11
Jul-Sept 11
0
0
0.0000
12
Okt-Des 11
1
1
0.0222
13
Jan-Mar 12
2
2
0.0455
14
Apr-Jun 12
2
2
0.0476
15
Jul-Sept12
0
0
0.0000
16
Okt-Des 12
3
3
0.0750
17
Jan-Mar 13
3
3
0.0811
18
Apr-Jun 13
2
2
0.0588
19
Jul-Sept 13
1
1
0.0313
20
Okt-Des 13
3
3
0.0968
Table 2. The Summary of Hazard Rate Value for Station 96741 (Jakarta/Tanjung Priok,
Indonesia).
No
Interval
N
Hazard Rate (T. Priok Station)
1
Jan-Mar 09
2
2
0.0333
2
Apr-Jun 09
0
0
0.0000
3
Jul-Sep 09
0
0
0.0000
4
Okt-Des 09
0
0
0.0000
5
Jan-Mar 10
1
1
0.0517
6
Apr-Jun 10
1
1
0.0175
7
Jul-Sept10
1
1
0.0179
8
Okt-Des 10
1
1
0.0364
9
Jan-Mar 11
2
2
0.0370
10
Apr-Jun 11
1
1
0.0385
11
Jul-Sept 11
0
0
0.0000
12
Okt-Des 11
0
0
0.0000
13
Jan-Mar 12
3
3
0.0392
14
Apr-Jun 12
0
0
0.0000
15
Jul-Sept12
0
0
0.0000
16
Okt-Des 12
2
2
0.0625
17
Jan-Mar 13
3
3
0.0652
18
Apr-Jun 13
1
1
0.0465
19
Jul-Sept 13
3
3
0.0238
20
Okt-Des 13
1
1
0.0732
3.3 Discussion
Based on case study, it appears that the hazard rate at each interval for two adjacent
stations is different. This value can be seen in Table 1 and Table 2. The same hazard
rate can be seen at interval of January until March 2009. This means that the hazard
rate at the beginning of the observation is not depend on the number of months that
have high rainfall. Although these two adjacent stations, but based on the results in
Table 1 and Table 2 shows that the value of the hazard rate for the same time interval
for both stations is different. This is consistent with that proposed by Lu Y and Qin
in 2012 [4] and Jien-Ji Tu in 2013 [10]. The difference is very reasonable because
based on Suroso in 2006 [5], in addition to the daily rainfall amount received in an
area, the intensity of rainfall in some areas also influenced by altitude. The lower the
altitude, the potential rainfall received will be more, because in general the lower the
temperature the higher regions. In addition, factors that affect the intensity of rainfall
in one place is the distance from the source of water (evaporation). The closer the
source of water, the higher the potential hujanya. Therefore, the value of the hazard
rate for the meteorological station of Tanjung Priok is lower than the meteorological
station Curug / Budiarto.
4. Acknowledgement
This study was supported by research “ Unggulan Perguruan Tinggi 2015 “ of
Hasanuddin University, Indonesia.
5. Conclusions
The result of hazard rate estimation of renewal point process with the inter event
time is exponentially distributed shows that hazard rate processes is constant, so the
rate of occurrence at given time t is not affected by the history of arrivals prior to t .
Application of the model to estimate the hazard rate shows that although
meteorological station Curug and Tanjung Priok is not so far, but they have the hazard
rate is not the same. Hazard rate value of Tanjung Priok is lower than the other.
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