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Home > Archives > 2013, Volume 47, Issue Number: 17
2013, Volume 47, Issue Number: 17
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Table of Contents
Articles
Generalized inequalities of Simpson-like type for functions whose derivatives in absolute value are
(alpha; m)-convex
Jaekeun Park 1-9
Determining the Number of Clusters by a Bayesian Approach
Degang Zhu 10-14
Pricing of European options using a cubic spline collocation method
A. Serghini, A. El hajaji, E.B. Mermri, K. Hilal 15-28
A smoothing algorithm to identify sharp discontinuities and peaks for a backward heat
conduction problem
Yixin Dou, Hengshan Hu, Bo Han 29-38
New Look for DHF Relative Risk Analysis Using Bayesian Poisson-Lognormal 2-Level Spatio-Temporal
Mukhsar , N. Iriawan, B. S. S. Ulama, Sutikno 39-47
On Image Reconstruction Algorithms with Discrete Radon Transform
Tanuja Srivastava, Nirmal Yadav 48-60
Existence and stability of anti-periodic solutions for an impulsive neural networks on time scales
Meng Hu 61-69
Smoothing of GRF Data Using Functional Data Analysis Technique
W. R. Wan Din, A. S. Rambely, A. A. Jemain 70-77
Almost periodic solution of recurrent neural networks with backward shift operators on time scales
Lili Wang, Meng Hu 78-86
Mathematical Simulation Model for Movement PDF
Trajectory of Backward Sliding Shot Put
YuHua Wu 87-94
Chebyshev wavelets method for solving Troesch’s problem
Changqing Yang, Jianhua Hou, Yan Xiong 95-103
Coordination and First-mover Advantage of Three-echelon Supply Chain
Yumei Hou, Fangfang Wei, Xin Tian, Lijun Ma 104-121
Generalized Method of Moments (GMM) Model for Financing Decision and Capital Structure on Manufacturing Enterprises’ Export Capacity
Guohua He, Deng Shang, Minwen Ye 122-130
Numerical Modeling for chaotic characteristics of oil pipeline pressure time series
Jianjun Xu, Shengnan Liu, Bin Xu, Xu Xu 131-139
Simulation and Video Analysis for Human Motion of Wushu Routine Teaching
Hai Yu, Yongbing Chen 140-147
TOPSIS Model and Grey Relational Analysis for the Football Evaluation
Dongjiao Huang 148-155
Data analysis approaches of incomplete fuzzy soft sets
Sisi Xia, Zhi Xiao, Xin Dang 156-168
Dynamically Predicting Reperforation Opportunity for Polymer Flooding
YIN Daiyin, DUAN Yingjiao 169-177
Statistics Analysis of Calibration Precision for Freehand Ultrasound Image
Yao Rao, Chen Minye, Xu Hairong, Cao Damin, Yao Peng
178-187
Mathematical Pricing Model with Dilution Effect and Firm-value Process Volatility for Bond with Attached
Warrant
Jie Miao 188-196
Numerical Simulation for Projectile Damped Motion in Basketball Movement Trajectory
Shiliang You 197-203
A Mathematical Model for Equity Financing Substitution Effects of Bio-pharmaceutical Listed
Companies
Wu Xiaogang, Wang Pengyuan, Du Rongwei 204-212
A Node Importance Evaluation Method for Complex Networks Based on M-order Neighbour Importance
Contribution
Zhang Xiping, Li Yongshu, Liu Gang, Wang Lei 213-221
Process Optimization of the Cold-Rolled Ribbed Steel Using GA and RBF Neural Network
Bangsheng Xing, Changlong Du 222-229
Analysis of Carbon Emission Based on Stochastic IPAT Model
Gang Du 230-238
Point Selection Model for a Railway Strategic Loading PDF
Xiaoping Guang, Liang Wu, Deyang Kong 239-246
Application of Coordination Control Optimization Method of Urban Main Road
Yihai Tian, , Qiong Wang, Lihong Yao, Xiaoping Guang
247-254
Numerical Simulation of H1N1 Virus Propagation Model Based on Small-world Network
Hong Wang, Ming Yang, Zhidan Lv, Zhaoguo Huang, Liang Wu, Junwei Zeng
255-261
Transient Fault Location for 10kv Distribution System Using Line Voltage and Zero-module Current
Qiao Zhanjun, Li Fuling, Li Yong 262-270
Finite Difference Analysis on the Generation of Heat by Spin Friction of the Projectile-Loaded Equipments
Shengliang Hu, Chao Mao 271-277
Application of Data Mining Technology in Analysis of Hierarchical Nursing Effects
Binbin Ji, Yujia Ren, Siyuan Tang 278-285
A Robust Edge Detection Algorithm Based on CR-DSmT
Kuixian Qiao 286-293
Chaotic Detection for Doppler Signal of Radio Fuze in Strong Noise
Xiaopeng Yan, Yongni Mou, Ping Li, Ruili Jia 294-301
Concept Association Mining Based on Clustering and Association Rules
Cuncun Wei 302-310
Land Cover Classification of High Resolution Images Using Superpixel-based Conditional Random Fields
Yun Yang 311-319
Dynamic Mechanism Analysis of Sustained Innovation in SMEs of Science and Technology
Wencai Cao, Miyuan Shan 320-329
Numerical Simulation of Combustion and Emission in Medium-Duty Diesel Engine Fuelled with Biodiesel
Blended Fisher-Tropsh Diesel
Zhifei Wu, Tie Wang, Ruiliang Zhang, Jianjun Zhu, Yonghui Deng
330-336
Multi-circle Detection Algorithm Based on Symmetry Property
Lianyuan Jiang, Peihe Tang, Yingjun Zhu, Jianbing Jiang, Yalan Zhang
337-345
Fast Texture Image Segmentation Algorithm Based on Dual-tree Complex Wavelet Transform
Yanli Hou 346-353
Virtual Fatigue and Durability Integrated Simulation Analysis of Rear Axle Housing Based on Mode
Superposition Method
Yiting Kang, Wenming Zhang, Yu Zhou 354-362
Bionic Flapping-Wing Mechanism Modelling and Simulation of Flapping Wingtip Trajectory
Zhaoxia He, Lan Liu, Xijin Zhang 363-371
An Algorithm for Parsing the Simple Semantic Units Based on Semantic Relevancies
Yuntong Liu, Jing Xiong 372-379
An Incremental Density Clustering Algorithm for Chaotic Time Series
Hui Li, Dechang Pi, Min Jiang 380-389
Short-term Power Load Forecasting Using Support Vector Machine based on Differential Evolution
Weiguo Zhao, Jianmin Hou, Gangzhu Pan, 390-398
Yanning Kang
Cost Analysis and Earning Allocation for Jointly Managed Inventory Based on One Supplier and Many
Producers
Xiaojuan Sheng, Xinzhong Bao, Zhe Wang 399-407
A Smoothing Algorithm with Momentum for Training Max-Min Fuzzy Neural Networks
Long Li, Rui Xiao, Guohui Zhang 408-415
Dynamical Behaviors of a Discrete SIR Epidemic Model with Nonmonotone Incidence Rate
Trija Fayeldi, Agus Suryanto, Agus Widodo 416-423
Approximate methods for a family of fractional differential equations
Jianhua Hou, Yan Xiong 424-430
Improved Faulty Line Detection Method for Small Current Grounding System
Bo Li 431-439
Minimizing of Shortfall Risk in a Jump-Diffusion Model with Continuous Dividends
Yunfeng Yang, Hao Jin 440-448
Risk Analysis and Accident Risk Assessment for the Aviation Sector
I. Üçkardeş, D. Ünal, N. Çaliş, Z. F. Antmen 449-461
New Bounds of Mutual Incoherence Property on Sparse Signals Recovery
Shiqing Wang, Limin Su 462-477
The Optimal Combination Model Building and Application of Linear Regression Based On Prediction
of Sports Scores
Wei Ye 478-485 ISSN: 0973-7545
New Look for DHF Relative Risk Analysis Using Bayesian Poisson-Lognormal 2-Level Spatio-Temporal
Mukhsar1, N. Iriawan2, B. S. S. Ulama2, and Sutikno2
1 Statistics Department, Institut Teknologi Sepuluh Nopember (ITS) Surabaya-Indonesia 60111; Mathematics Department Haluoleo University
Kendari-Indonesia 93231 Email: [email protected]; [email protected]
2Statistics Department, Institut Teknologi Sepuluh Nopember (ITS)
Surabaya-Indonesia 60111 Email: [email protected]; [email protected]; [email protected]
ABSTRACT
Spatial convolution (Poisson-Lognormal) model with Bayesian approach is developed into spatio-temporal form by adding temporal trend to analyze DHF relative risk. The DHF data had been considered as a 2-level hierarchy phenomenon. The developed model is divided into two models, e.i. Bayesian Poisson-Lognormal 2-level (BP2L) spatio-temporal due to the spatial random effects and extended of BP2L (EoBP2L) spatio-temporal due to the spatio-temporal random effects. The works of the models were demonstrated by using MCMC Gibbs sampler to analyze DHF data on 31 districts in Surabaya city during 120 months (2001-2010) using covariate such as temperature, humidity, rainfall, and population density. Based on virtue of relative risk visualization, MC error, and deviance, the EoBP2L spatio-temporal not only has better performance than the BP2L, but it also break up Surabaya city into two zones of DHF hot spot; Sawahan and Tambaksari district. January is the best time for DHF intervention every year in both hot spots.
Keywords: Bayesian spatio-temporal, Convolution, Deviance, MCMC Gibbs sampler, DHF, Poisson-Lognormal, Random effects
Mathematics Subject Classification: 62-07, 62F15, 65C05, 62P12, 62P99
1. INTRODUCTION
Dengue Hemorrhagic Fever (DHF) cases always threaten the dense population every year in
Indonesia because of its tropical climate. Mukhsar, et al. (2012) had applied the Bayesian spatial
convolution (Poisson-Lognormal) model for analyzing the DHF relative risk using DHF data in
Surabaya for 31 districts on 2010. Chowell, et al. (2011) had stated that the DHF case is not only a
spatial-dependent but also a temporal-dependent observable fact. Furthermore, Iriawan, et al. (2012)
considered the DHF data is structured as 2-level where the district is nested to the city as sub-level.
This paper introduces the development of the Poisson-Lognormal model in 2-level spatio-temporal
form by adding trend temporal. There are two models that would be developed based on setting of the
random effects, which is varying spatially and varying spatio-temporally. The first model would be
called Bayesian Poisson-Lognormal 2-level (BP2L) and the second model would be called extended of
BP2L (EoBP2L) spatio-temporal. The parameters of both models were estimated by using MCMC
International Journal of Applied Mathematics and Statistics,Int. J. Appl. Math. Stat.; Vol. 47; Issue No. 17; Year 2013, ISSN 0973-1377 (Print), ISSN 0973-7545 (Online) Copyright © 2013 by CESER Publications
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Gibbs sampler (Rafida, et al., 2010; Astutik, et al., 2013; Fithriasari, et al., 2013) based on its full
conditional distributions (Mukhsar, et al., 2013). Both models were employed to analyze DHF data on
31 districts in Surabaya city during 120 months (2001-2010) and result would be compared. Finally,
the better model would be used to analyze the relative risk of the DHF in Surabaya city.
2. BAYESIAN POISSON-LOGNORMAL 2-LEVEL SPATIO-TEMPORAL
The BP2L spatio-temporal model is constructed based on the characteristics of the DHF data
(Mukhsar, et al., 2013). This developed model structure is imbued by the model that was previously
presented by Eckert, et al. (2007); Neyens, et al. (2011). The random effects of the BP2L spatio-
temporal are spatially dependent.
Suppose the DHF count sty is identically distributed Poisson with parameter st� . Poisson variability is
influenced by st� that depends on district s at time t, expressed as
T0
1
~ ( )
exp ( ) , 1, , , 1, , , 1,..., ,
st st st
P
st st p pst s s s zp
y Poisson
e x u v t s S t T p P
� �
� � � � ��
� � � � �� � � � � � � � �� � �� �� �
� � � (1)
where S is the number of districts, T is length of time observation, P is the number of covariates, ste is
an expected count in district sth at time tth, pstx is pth covariate in district sth at time tth, su is
uncorrelated random effect at district sth, sv is correlated random effect (CAR model) at district sth, and
( )s zt� �� is trend temporal.
Full conditional distributions of each parameter in BP2L spatio-temporal had been derived by Mukhsar,
et al. (2013), as follow
� ( )mp� is generated for m times iterations from
2(0) (0)2 211 11(0)12 12
, exp ,A A
p A AN� �� �
�� �
� � �� � � � � �� �� �� �
� (2)
where � �2(0)T T11 12
1 1 1 1 1 1, 1 ,
T S P T S P
st st pst st pstt s p t s p
A y e x A e x��� � � � � �
� �� � � � � �� � � �
�� � �� � � (0)�� is initial value.
� ( )msu is generated for m times iterations from
� �2(0)22 (0) 1313
14 14~ , exp ,
S AAu us uA Au N �� �
� �� � � � � � �� �� �� �
(3)
where � � � �(0)13 14
1 1 1 1, 1 ,
T S T S
st st u stt s t s
A y e A e�� � � �
� � � ��� �� �and (0)u� is initial value.
� ( )msv is generated for m times iterations from
International Journal of Applied Mathematics and Statistics
40
� �2(0) (0)2 2(0)15 152
16 16~ , exp ,
SA Av v
s v sA Av N D� ��� � � �� �� � � �� � �� �� �� �
(4)
where � �2(0) (0) (0)15 16
1 1 ( ) 1 1, ,
T S S T S
st st v j v st v st s j s t s
A y e v A e D�
�� � �� � � � �
�� � � � � � � � �� ��� � �� (0)
v� is initial value.
� ( )m� is generated for m times iterations from
2(0) (0)2 2(0)17 1718 18
, exp ,A AA AN � �� �
�� � � � �� �� � � �� � �� �� �� �
� (5)
where � � � �(0)17 18
1 1 1 1, 1 ,
T S T S
st st stt s t s
A y e A e ��� � � �
�� � � � �� �
�� �� �and (0)�� is initial value.
� ( )ms� is generated for m times iterations from
2(0 ) (0 ) 2(0 ) (0 )22 219 19( )1
(0 ) 220 20~ , exp ,
SSjA Aj s
s A ADsDsN
� � ��� ��� ���
��
�
� � �� � � �� � � � � � �� � �� � � � �� � � �� � � � � �� � � �� �� �� �� �
� (6)
where � �2(0) (0) (0) (0)19 20
1 1 ( ) 1 1, ,
T S S T S
st st st sjt s j s t s
A y e A e D� � ��
� � � � �� � � � �
�� � � � � � � � �� ��� � �� � and (0)
��
is initial value.
3. THE EXTENDED OF BP2L SPATIO-TEMPORAL
The Extended of BP2L (EoBP2L) spatio-temporal is expanded from model (1) which is focused on the
modification of the uncorrelated heterogeneity spatially su and the correlated heterogeneity spatially
sv into the uncorrelated heterogeneity of spatio-temporal, stu , and the correlated heterogeneity of
spatio-temporal, stv ,
T0
1
~ ( )
exp ( ) , 1, , , 1, , , 1,..., .
st st st
P
st st p pst st st s zp
y Poisson
e x u v t s S t T p P
� �
� � � � ��
� � � � �� � � � � � � � �� � �� �� �
� � � (7)
Full conditional distributions of each parameter in the EoBP2L spatio-temporal could be found in
similar way as in BP2L spatio-temporal. Full conditional distributions for stu and s tv are as follow
� ( )mstu is generated for m times iterations from
� �2(0) (0)2 2(0)23 23
24 24~ , exp ,
STA Au ust uA Au N � ��
� � �� �� � � �� � �� �� �� �� �
(8)
International Journal of Applied Mathematics and Statistics
41
where � � � �(0)23 24
1 1 1 1, 1 ,
T S T S
st st u stt s t s
A y e A e�� � � �
� � � ��� �� and (0)u� is initial value.
� ( )mstv is generated for m times iterations from
2(0) (0)2 2(0)25 252
26 26~ , exp ,
STA Av v
st v sA Av N D� ��� � � �� �� � � � � �� � � � �� �� �� �
(9)
where � �2(0) (0) (0)25 26
1 1 ( ) 1 1, ,
T S S T S
st st v jt v s vt s j s t s
A y e v A D�
�� � �� � � � �
�� � � � � � � � �� ��� � �� (0)
v� is initial value.
4. RESULTS
4.1. Parameter Estimation
The BP2L and EoBP2L spatio-temporal are implemented for analyzing the DHF data on 31 districts in
Surabaya during 120 months (2001-2010). The covariates used in this case are humidity ( 1( )stX ),
temperature ( 2( )stX ), rainfall ( 3( )stX ), and population density ( 4( )stX ).
Gibbs sampler is used to estimate the parameters numerically by generating their values based on
their full conditional distributions (Congdon, 2010; Iriawan et al., 2010; Rafida, et al., 2010; Astutik, et
al., 2013; Fithriasari, et al., 2013) as stated in section 2 and section 3 for m time iterations. The BP2L
and the EoBP2L spatio-temporal have been run for 10000 iterations after discarding an additional
50000 iterations as burn-in, respectively. The convergence can be investigated by exploring the history
process in the same zone (Best and Elliott, 2004; Gelman et al., 2004; Gamerman and Lopes, 2006;
Ntzoufras, 2009). For example, the generated sample of parameter 3� related with rainfall ( 3( )stX ) of
BP2L and EoBP2L spatio-temporal were shown in Figure 1, while their densities are expressed in
Figure 2.
beta3
iteration50000 55000 6000
-3.0E-4-2.0E-4-1.0E-4
1.00E-42.00E-4
beta3
iteration50000 55000 6000
0.02.00E-44.00E-46.00E-48.00E-4 0.001
�
(a) (b)
Figure 1. History plot of generated sample of 3� for 10000 iterations after discarding an additional 50000 iterations as burn-in, (a) BP2L spatio-temporal, (b) EoBP2L spatio-temporal
Markov chain (MC) error of WinBUGS is determined upon window estimator ( w ) from autocorrelation
variance sample (Gelman and Hill, 2007; Lawson, 2008; Marin and Robert, 2007; Eberley and Carlin,
2000)
International Journal of Applied Mathematics and Statistics
42
� �� �� �� �� � �w
kk=1
SD G(�) ˆMCerror G(�) = 1+ 2 � G(�)m
. (10)
where � �k�̂ G(�) is autocorrelation estimation at lag k. The posterior summary for BP2L and EoBP2L
spatio-temporal are shown in Table 1.
beta3 sample: 10001
-4.0E-4 -2.0E-4 0.0
0.0 2500.05.00E+3 7500.01.00E+4
beta3 sample: 10001
-5.0E-4 0.0 5.00E-4
0.01.00E+32.00E+33.00E+34.00E+3
�
(a) (b)
Figure 2. Density plot of 3� for 10000 iterations after discarding an additional 50000 iterations as burn-in, (a) BP2L spatio-temporal, (b) EoBP2L spatio-temporal
Table 1: Posterior summary of PB2L spatio-temporal and EoBP2L spatio-temporal for 10000 iterations
after discarding an additional 50000 iterations as burn-in, respectively
Node Mean SD MC error 2,50% Median 97,50% BP2L spatio-temporal beta0 - 0.3358 0.3097 0.0286 - 0.9124 - 0.348 0.3545 beta1 0.002532 0.001167 6.98E-05 2.17E-04 0.002538 0.004729 beta2 0.005005 0.00718 6.47E-04 - 0.00966 0.00492 0.01798 beta3 - 4.52E-05 4.38E-05 1.20E-06 - 1.3E-04 - 4.48E-05 3.99E-05 beta4 - 9.04E-04 0.001079 9.34E-05 - 0.0034 - 8.37E-04 0.001204 deviance 12950 EoBP2L spatio-temporal beta0 - 0.2138 0.1208 0.02026 - 0.4704 - 0.2349 - 0.0141 beta1 0.001071 0.00233 1.9E-04 - 0.00375 0.001276 0.005204 beta2 - 0.0015 0.00550 4.27E-04 - 0.01313 - 0.00126 0.00902 beta3 4.75E-04 1.15E-04 3.97E-06 2.49E-04 4.75E-04 7.05E-04 beta4 0.001435 3.85E-04 1.72E-05 6.99E-04 0.001425 0.002232 deviance 8319
4.2. Model Performance and Relative Risk Interpretation
Table 1 shows that all of the covariates in the BP2L spatio-temporal are not statistically significant to
influence the DHF case, except the humidity factor. While in real life phenomena, the rainfall and
population density affect to increase of DHF case. Whereas, the EoBP2L spatio-temporal shows the
right identification. Those facts are shown by increasing rainfall (indicated to the positive value of 3� )
and population density (indicated to the positive value of 4� )�support to increase of DHF case. This
regard conforms to the real conditions, which the DHF cases are always attacking in the dense
population along with the increasing of the rainfall. Moreover, the deviance of EoBP2L spatio-temporal
is smaller than BP2L spatio-temporal. This result shows that the EoBP2L spatio-temporal is more
appropriate model for analyzing the DHF relative risk (RR) in Surabaya city, than BP2L spatio-
temporal. Based on (7), the RR pattern can be seen as follows
International Journal of Applied Mathematics and Statistics
43
4T
01
RR exp ( ) , 1, ,31, 1,...,120.st p pst st st s zp
x u v t s t� � � ��
� � � � �� � � � � � � �� � �� �� �
� � (11)
The compliance pattern of RR and observation is demonstrated in Figure 3.
0 12 25 37 49 61 72 85 97 110Time (month)
Obs.
EoBP2L Spatio-Temporal
Figure 3. The compliance RR pattern of EoBP2L spatio-temporal and data observation of DHF in Surabaya city in 31 districts during 120 months
Figure 3 shown nine highest DHF cases. Those cases, timing and location are shown in Table 2. For
illustration only, the mapping of highest case on January 2006 is further discussed in Figure 4.
Figure 4. Mapping views of highest risk of Highest DHF in each district in Surabaya city on January 2006
Table 2: Location of each nine highest DHF cases
DHF period of infection District of DHF highest relative risk December 2001 Sawahan and Tambaksari district January 2003 Sawahan, Tambaksari, and Wonokromo district January 2004 Sawahan and Wonokromo district January2005 Sawahan, Tambaksari, and Wonokromo district January 2006 Sawahan, Tambaksari, Wonokromo, Simokerto district December 2006 Sawahan and Tambaksari district January 2008 Sawahan, Tambaksari, Wonokromo, and Simokerto district January 2009 Sawahan, Tambaksari, Wonokromo, and Simokerto district February 2010 Sawahan, Tambaksari, Wonokromo, and Simokerto district
Table 2 shows that the Sawahan and Tambaksari districts are consistent location as the highest DHF
cases in Surabaya city. Sawahan district is hot spot in zone 1 and Tambaksari district is hot spot in
zone 2 (Figure 5).
International Journal of Applied Mathematics and Statistics
44
N
Figure 5. Mapping zone of the hot spots of DHF risk in Surabaya city
5. CONCLUSION
This paper have already demonstrated the work using MCMC Gibbs sampler in WinBUGS of the BP2L
spatio-temporal compared with the EoBP2L spatio-temporal applied for DHF data in 31 districts in
Surabaya city, Indonesia, during 120 months (2001-2010). The result shows that all of the covariates
in the BP2L spatio-temporal are not statistically significant to influence the DHF case, except the
humidity factor. While in real life phenomena, the rainfall and population density affect to increase of
DHF case. Whereas, the EoBP2L spatio-temporal shows the right identification, except the
temperature factor ( 2� ) is not statistically significant. Those facts are shown by increasing humidity
(indicated to the positive value of 1� ), rainfall (indicated to the positive value of 3� ), and population
density (indicated to the positive value of 4� )�support to increase of DHF case. Moreover, the deviance
of EoBP2L spatio-temporal is smaller than BP2L spatio-temporal. Furthermore, the EoBP2L spatio-
temporal is better than BP2L spatio-temporal model to analyze the DHF risk in Surabaya city.
Surabaya city could be assumed to have two zones as the hot spot of the DHF risk. Sawahan district
is the hot spot in zone 1 and Tambaksari is the hot spot in zone 2. Every January, therefore, is an
appropriate time for the DHF case intervention in Sawahan and Tambaksari.
6. ACKNOWLEDGEMENTS
This article is a part of Laboratory’s research grant and doctoral research at Statistics Department of Institut Teknologi Sepuluh Nopember (ITS), Surabaya, Indonesia, granted by LPPM Institut Teknologi Sepuluh Nopember (ITS), number 1027.116/IT2.7.PN.01/ 2012. We thank Head of BPS and BMKG Surabaya city.
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