Spatio Temporal Modelling for Government Policy the COVID-19 Pandemic in East Java


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(4) (2021), Pages 218-226 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: November 07, 2020 Reviewed: December 03, 2020 Accepted: April 12, 2021 
DOI: http://dx.doi.org/10.18860/ca.v6i4.10639     

Spatio Temporal Modelling for Government Policy the COVID-19 
Pandemic in East Java 

Atiek Iriany1, Novi Nur Aini1, Agus Dwi Sulistyono2  

1 Department of Statistics Faculty of Mathematics and Natural Sciences, Brawijaya 
University, Indonesia 

2 Faculty of Fisheries and Marine Science, Brawijaya University, Indonesia 

 

Email: atiekiriany@ub.ac.id 

ABSTRACT  

COVID-19 has cursorily spread globally. Just in four months, its status altered into a pandemic. In 
Indonesia, the virus epicenter is identified in Java. The first positive case was identified in West 
Java and later spread in all Java. The Large-scale Social Restrictions are seemingly inefficient as 
the SARS-CoV-2 transmission remains. As such, the government is struggling to find anticipatory 
policies and steps best to mitigate the transmission. In this particular article, we used a Spatio-
temporal model method for the total COVID-19 cases in Java and forecasted the total cases for the 
next 14 days, allowing the stakeholders to make more effective policies. The data we were using 
was the daily data of the cumulative number of COVID-19 cases taken from www.covid19.go.id. 
Data modeling was conducted using a generalized Spatio-temporal autoregressive model. The 
model acquired to model the COVID-19 cases in Java was the GSTAR(1)(1,0,0) model. 
 
Keywords: COVID-19; forecasting, pandemic; spatio-temporal 

INTRODUCTION  

As stipulated by WHO on 12 March 2020, COVID-19 had become a pandemic [1]. 
The virus, firstly identified in Wuhan in December 2019, rapidly spread throughout China 
and other 190 countries [2]. No research exactly explains how the SARS-CoV-2 was 
initially transmitted, but, in the meantime, it is believed that humans transmit this virus 
to humans. Later research reveals that symptomatic patients transmit SARS-CoV-2 
through droplets or sneezes [3]. Moreover, another research mentions that SARS-CoV-2 
can live in gas particles, e.g., air (generated through nebulizer) for approximately three 
hours [4]. Due to its relatively rapid transmission and mortality rate which cannot be 
overlooked and no definitive therapy found, COVID-19 is one of the diseases to which we 
should alert [5].  

The Coronavirus epicenter in Indonesia is identified in Java. The first positive case 
was identified in West Java and later spread in all Java. It indicates that adjacent locations 
closely pertain to the SARS-CoV-2 transmission. In response to the virus, China’s social 
distancing regulation is proven effective to stabilize the virus transmission, and hence the 
number declines [6]. Indonesia, similar to China, issues the same regulation, namely the 
Large-scale Social Restrictions (PSBB). Nevertheless, the regulation is seemingly 
inefficient as the SARS-CoV-2 transmission remains. As such, the government is struggling 
to find anticipatory policies and steps best to mitigate the transmission. 

http://dx.doi.org/10.18860/ca.v6i4.10639
http://www.covid19.go.id/


Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java 

Atiek Iriany 219 

Many researchers, e.g., Jia et al. [7], Albana [8], and Fajar [9] have studied the 
COVID-19 transmission and aim to recommend some anticipatory efforts. Meanwhile, we 
made COVID-19 modeling using a Spatio-temporal approach due to the SARS-CoV-2 
transmission, which is mostly influenced by interplay, and numerous positive cases. 
Several researchers used the Spatio-temporal model [10] and [11]. One of the methods 
used to handle data attributed to time and location was Generalized Space-time 
Autoregressive (GSTAR). Some researchers, such as Iriany [12], Ruchjana [13], and 
Prastyo [14], prefer this method. In this particular article, we used a Spatio-temporal 
model for the total COVID-19 cases in Java and forecasted the total cases for the next 14 
days, allowing the stakeholders to make more effective policies. 

 

METHODS  

Data Source 

The data we were using in this research were the daily data of the cumulative 
number of COVID-19 cases taken from www.covid19.go.id.  

 
Data Stationarity 

According to the stationary time series data, neither a sharp decrease nor an 
increase in data value nor fluctuated data was found around the constant mean value [15]. 
Stationary data had the mean 𝐸(𝑍𝑡) = µ and variance 𝑉𝑎𝑟(𝑍𝑡) = σ2. The mean value 
conditioned that data had to be stationary, so neither decrease nor an increase in data 
from time to time was allowed [16]. Furthermore, the characteristic of a stationary time 
series was endlessly constant average and variance. There were two types of time series 
stationarity, namely stationarity to variance and the mean. 
a. Stationarity to Variance 

Stationarity to variance was if 𝑉𝑎𝑟(𝑍𝑡) = 𝑉𝑎𝑟(𝑍𝑡−𝑘) for all t and k, the variance 
was constant from time to time [17]. To observe whether or not the data was stationary 
to variance, we used a Box-Cox plot. Non-stationary data could be altered into stationary 
ones through transformation. 
b. Stationarity to the Mean  

Stationarity to the mean was if 𝐸(𝑍𝑡) = 𝐸(𝑍𝑡−𝑘) for all t and k, the mean function 
remained constant from time to time. Stationarity to the mean was observed using the 
ACF (Autocorrelation Function) plot or the Dickey-Fuller test. Non-stationary data could 
be altered into stationary ones through differencing. 
Generalized Space-Time Autoregressive Integrated (GSTAR) 

The AR order was determined using the MPACF plot. Correlation between Zt and 
Zt+k, after a dependence relationship, was linear. The variables Zt+1, Zt+2, …, and Zt+k-1 were 
thus negated. The formula of correlation partial matrix function is as follows: 
 

ϕkk = 
𝑐𝑜𝑣 [(𝑍𝑡−�̂�𝑡),(𝑍𝑡+𝑘−�̂�𝑡+𝑘)]

√𝑣𝑎𝑟(𝑍𝑡−�̂�𝑡)√𝑣𝑎𝑟(𝑍𝑡+𝑘−�̂�𝑡+𝑘)
        (1) 

Where 
ϕkk = Partial correlation matrix coefficient at lag k 
𝑍𝑡 = Observation data at the time t 
�̂�𝑡  = Predictor for 𝑍𝑡 
𝑍𝑡+𝑘  = Observation data at the time 𝑡 + 𝑘 
�̂�𝑡+𝑘  = Predictor for 𝑍𝑡+𝑘  

http://www.covid19.go.id/


Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java 

Atiek Iriany 220 

The partial autoregression matrix at lag s became the last matrix coefficient when 
the data were leveraged for the vector autoregression process of the order s.  

The best model was selected among some models considered feasible for MPACF 
testing. Model selection was conducted using AIC. The less the AIC value in a model, the 
better the model. The quantification of the AIC value was as follows: 

 

𝐴𝐼𝐶(𝑖) = ln (|𝑆(𝑝)| +
2𝑝𝑏2

𝑇
)        (2)    

Where: 
b  = the number of predicted parameters in the model 
T = the number of observations 
S(p)  = residual sum of squares 
p = VAR model order 
 

The GSTAR model was introduced by Borovkova, Lopuha, and Ruchjana in 2020 in 
Wutsqa et al. [18]. It was more flexible and generalized than the STAR model and did not 
require the same parameter values at all locations. The GSTAR model (𝑝, 𝜆1, … . , 𝜆𝑙)  is 
written as follows [19]: 

 
Zt = ∑ [Φ𝑘0 + 

𝑝
𝑘=1 Φ𝑘1𝑊] 𝑍𝑡−𝑝 +  𝑒𝑡                      (3) 

Where: 
Φ𝑘0 = diag (𝜙𝑘0

1  , … , 𝜙𝑘0
𝑛 ), diagonal matrix of the parameter space-time lag spatial 0 and  

   the parameter autoregressive lag at the time kth 
Φ𝑘1  = diag (𝜙𝑘1

1  , … , 𝜙𝑘1
𝑛 ), diagonal matrix of the parameter space-time lag spatial 1 and  

   the parameter autoregressive lag at the time kth  

W = weighing matrix (N×N) selected as such that 𝑊
𝑖𝑖
(𝑘)

= 0 dan  ∑ 𝑊
𝑖𝑗
(𝑘)

= 1𝑖≠𝑗  

e(t) = the white-nose vector in size of (N × 1) 
Z(t)  = the random vector in size of (N × 1) at the time t 
 

Suhartono and Subanar [20] introduced a new method for determining weight 
using the result of cross-correlation normalization between locations at a congruent time 
lag.  

 

�̂�𝑖𝑗(𝑘) =  𝑟𝑖𝑗 (𝑘) = 
∑ [𝑍𝑖(𝑡)− 𝑍𝑙̅̅ ̅]

𝑛
𝑘+1  [[𝑍𝑗(𝑡−𝑘)− 𝑍𝑗]

̅̅ ̅̅  

√(∑ [𝑍𝑖(𝑡)− 𝑍𝑙̅̅ ̅]
2𝑛

𝑡=1 )(∑ [𝑍𝑗(𝑡)− 𝑍𝑗
̅̅ ̅]2𝑛𝑡=1    

                   (4) 

 
The determination of location weight for the GSTAR model (1;p) is as follows: 
 

 wij = 
𝑟𝑖𝑗(1)

∑ |𝑟𝑖𝑘(1)|𝑘≠1
          (5) 

 

with i ≠ j and the weight had fulfilled ∑ 𝑤𝑖𝑗𝑖≠𝑗  = 1. 

The weight of cross-correlation normalization represented the variance of 
correlation between locations occurring in the data. 
  



Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java 

Atiek Iriany 221 

RESULTS AND DISCUSSION  

The COVID-19 cases in Indonesia were ever-increasing, and Java was regarded as 
the transmission epicenter. The increase in the COVID-19 cases is depicted in Figure 1.  
 

 

 
Figure 1. The plot of the time series of COVID-19 cases in each province 

 
Figure 1 indicates that as of 2 March-18 May 2020, the COVID-19 cases increased 

in all provinces in Java. On 18 May 2020, the highest number of cases, 5,555, was 
reportedly in Jakarta, whereas the lowest one, 185, was in Yogyakarta. Using the data of 
the total COVID-19 cases in six provinces in Java, we identified the correlation between 
provinces and the COVID-19 transmission in Java. Correlation between locations was 
identified using Pearson’s correlation between provinces. The result of Pearson 
correlation quantification is presented in Table 1. 

Table 1. The Correlation Value of the COVID-19 Cases between Provinces in Java 

  
Banten Jakarta West Java 

Central 
Java 

Yogyakarta East Java 

Banten 1 0.994 0.994 0.982 0.981 0.973 
Jakarta 0.994 1 0.995 0.989 0.975 0.967 
West Java 0.994 0.995 1 0.992 0.986 0.977 
Central Java 0.982 0.989 0.992 1 0.983 0.980 
Yogyakarta 0.981 0.975 0.986 0.983 1 0.996 
East Java 0.973 0.967 0.977 0.980 0.996 1 

 
In Table 1, we can see that the data of the number of the COVID-19 cases in six 

provinces in Java had a high Pearson’s correlation value which was higher than 0.9. It 
implies that the correlation of the COVID-19 cases between provinces in Java was strong. 

 
Data Stationarity Test 

Data stationarity testing was performed in two stages which were stationarity to 
variance and stationarity to the mean. Stationarity to variance was tested using the box-
cox transformation. Data were regarded stationary if the lambda value was 1, signifying 
that Var(Zt) = Var(Zt-k). The result of the stationarity test to variance is shown in Table 2. 

Table 2. The Result of Box-Cox Transformation 

Location λ 
Transformation Final Transformation 

Trans. λ Trans. λ  
Banten 0.20 Zt0.20 1.00 - - Zt0.20 
Jakarta 0.20 Zt0.20 1.00 - - Zt0.20 



Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java 

Atiek Iriany 222 

Location λ 
Transformation Final Transformation 

Trans. λ Trans. λ  
West Java 0.19 Zt0.19 1.00 - - Zt0.19 
Central Java 0.00 Ln(Zt) 1.00 - - Ln(Zt) 
Yogyakarta 0.00 Ln(Zt) 0.00 Ln(Zt) 1.00 Ln(Ln(Zt)) 
East Java 0.00 Ln(Zt) 0.50 Zt0.50 1.00 Ln(Zt)0.50 

As seen in Table 2, the initial data had not fulfilled the stationarity to variance yet. 
Several transformations were thus called for. After conducting the data stationarity test, 
we did the stationarity test to the mean. The test was conducted using an augmented 
Dickey-Fuller test. The result of the stationarity test to the mean is indicated in Table 3. 
 

Table 3. The Result of the Augmented Dickey-Fuller Test 

Location Lag 0 1 2 

Banten 
π 98.73 60.83 34.96 

p-value 0.001 0.001 0.001 

Jakarta 
π 107.89 32.98 21.15 

p-value 0.001 0.001 0.001 

West Java 
π 100.74 40.22 28.78 

p-value 0.001 0.001 0.001 

Central Java 
π 122.94 55.84 29.66 

p-value 0.001 0.001 0.001 

Yogyakarta 
π 75.55 51.84 42.09 

p-value 0.001 0.001 0.001 

East Java 
π 155.97 57.49 25.85 

p-value 0.001 0.001 0.001 

 
From the augmented Dickey-Fuller test, we acquired predicted values less than the 

real ones (0.05). It indicates that the data had fulfilled the stationarity to variance. 
 

Interpretation of the GSTAR Model Parameter 
Model identification was aimed to find the autoregressive GSTAR model order. The 

order was elicited by identification using AIC. The lag with the smallest AIC value was 
regarded as the autoregressive GSTAR model order. Table 4 lists the AIC values. 
  

Table 4. The AIC Value in Model Order Selection 

Lag MA 0 MA 1 MA 2 MA 3 MA 4 MA 5 

AR 0 34.0876 35.2909 35.5546 36.0924 36.8882 36.1828 

AR 1 31.9424 32.9755 33.3945 33.9363 33.9725 33.0619 

AR 2 32.0815 33.1868 33.0648 33.8555 34.487 34.3527 

AR 3 32.006 32.8989 33.096 35.128 35.9621 36.0125 

AR 4 32.9289 34.1187 32.882 35.2756 35.429 42.0827 

AR 5 34.6863 35.2259 35.61 35.7834 42.3544  

 
Table 4 shows the smallest AIC value at the lag AR(1) and MA(0), hence the 

GSTAR(1)(1,0,0) model.  
 

  



Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java 

Atiek Iriany 223 

Interpretation of the GSTAR Model Parameter 
The GSTAR model was a particular form of VAR engaging spatial elements. 

Estimating the GSTAR(1) (1,0,0) spatial parameters with the Ordinary Least Square 
method using cross-correlation normalization weight generated the following 
parameters. 

 
Table 5. The Parameters of the GSTAR(1)(1,0,0) Model 

Location Parameter Estimation 

Banten 
∅10

(1)
 1.015 

∅11
(1)

 0.793 

Jakarta 
∅10

(2)
 0.915 

∅11
(2)

 0.984 

West Java 
∅10

(3)
 0.758 

∅11
(3)

 1.031 

Central Java 
∅10

(4)
 -0.003 

∅11
(4)

 0.256 

Yogyakarta 
∅10

(5)
 0.118 

∅11
(5)

 0.061 

East Java 
∅10

(6)
 0.088 

∅11
(6)

 -0.013 

 
Referring to Table 5, we generated the matrix equation of the GSTAR(1)(1,0,0) 

model, which is as follows: 
 

[
 
 
 
 
 
 
𝑍1(𝑡)

𝑍2(𝑡)

𝑍3(𝑡)

𝑍4(𝑡)

𝑍5(𝑡)

𝑍6(𝑡)]
 
 
 
 
 
 

=  

[
 
 
 
 
 
1.015 0 0 0 0 0

0 0.915 0 0 0 0
0 0 0.758 0 0 0
0 0 0 −0.03 0 0
0 0 0 0 0.118 0
0 0 0 0 0 0.088]

 
 
 
 
 

[
 
 
 
 
 
 
𝑍1(𝑡 − 1)

𝑍2(𝑡 − 1)

𝑍3(𝑡 − 1)

𝑍4(𝑡 − 1)

𝑍5(𝑡 − 1)

𝑍6(𝑡 − 1)]
 
 
 
 
 
 

 

+  

[
 
 
 
 
 
0.793 0 0 0 0 0

0 0.984 0 0 0 0
0 0 1.031 0 0 0
0 0 0 0.256 0 0
0 0 0 0 0.061 0
0 0 0 0 0 −0.013]

 
 
 
 
 

 

[
 
 
 
 
 

0 0.256 0.116 0.209 0.183 0.236
0.205 0 0.187 0.217 0.160 0.231
0.192 0.223 0 0.233 0.150 0.201
0.092 0.273 0.194 0 0.231 0.210
0.156 0.242 0.079 0.276 0 0.247
0.134 0.241 0.180 0.124 0.321 0 ]

 
 
 
 
 

 

[
 
 
 
 
 
 
𝑍1(𝑡 − 1)

𝑍2(𝑡 − 1)

𝑍3(𝑡 − 1)

𝑍4(𝑡 − 1)

𝑍5(𝑡 − 1)

𝑍6(𝑡 − 1)]
 
 
 
 
 
 

+

[
 
 
 
 
 
 
𝑒1(𝑡)

𝑒2(𝑡)

𝑒3(𝑡)

𝑒4(𝑡)

𝑒5(𝑡)

𝑒6(𝑡)]
 
 
 
 
 
 

  

 
The following matrix equation was derived from the above equation. 
 



Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java 

Atiek Iriany 224 

[
 
 
 
 
 
 
𝑍1(𝑡)

𝑍2(𝑡)

𝑍3(𝑡)

𝑍4(𝑡)

𝑍5(𝑡)

𝑍6(𝑡)]
 
 
 
 
 
 

=  

[
 
 
 
 
 
1.015 0 0 0 0 0

0 0.915 0 0 0 0
0 0 0.758 0 0 0
0 0 0 −0.03 0 0
0 0 0 0 0.118 0
0 0 0 0 0 0.088]

 
 
 
 
 

[
 
 
 
 
 
 
𝑍1(𝑡 − 1)

𝑍2(𝑡 − 1)

𝑍3(𝑡 − 1)

𝑍4(𝑡 − 1)

𝑍5(𝑡 − 1)

𝑍6(𝑡 − 1)]
 
 
 
 
 
 

 

+  

[
 
 
 
 
 

0 0.259 0.118 0.212 0.186 0.239
0.202 0 0.184 0.213 0.157 0.227
0.198 0.229 0 0.240 0.155 0.207
0.024 0.069 0.049 0 0.059 0.054
0.009 0.015 0.005 0.017 0 0.015

−0.002 −0.004 −0.002 −0.002 −0.004 0 ]
 
 
 
 
 

 

[
 
 
 
 
 
 
𝑍1(𝑡 − 1)

𝑍2(𝑡 − 1)

𝑍3(𝑡 − 1)

𝑍4(𝑡 − 1)

𝑍5(𝑡 − 1)

𝑍6(𝑡 − 1)]
 
 
 
 
 
 

 

+

[
 
 
 
 
 
 
𝑒1(𝑡)

𝑒2(𝑡)

𝑒3(𝑡)

𝑒4(𝑡)

𝑒5(𝑡)

𝑒6(𝑡)]
 
 
 
 
 
 

 

 
From the model generated, we made a comparison between the actual and 

predicted data, in which we acquired an RMSE and MAPE value of 0.005 and 1.43, 
respectively. The two gave us a hint that the model generated was good. 

 
Prediction Result 

From the equation, we forecasted the total cases for the next 14 days, namely 19 
May-1 June 2020, the result of which is presented in Table 6. 
 

Table 6. The Predicted COVID-19 Cases on 19 May-1 June 2020 

 Banten Jakarta West Java Central Java Yogyakarta East Java 
1 628 5662 1689 1214 195 2321 
2 651 5738 1746 1249 201 2438 
3 674 5802 1804 1283 207 2561 
4 699 5850 1862 1318 214 2689 
5 725 5881 1920 1352 220 2822 
6 754 5892 1978 1386 227 2961 
7 784 5880 2035 1419 233 3105 
8 817 5841 2092 1450 240 3255 
9 852 5773 2148 1481 247 3411 

10 891 5672 2202 1509 254 3573 
11 933 5533 2254 1535 260 3741 
12 979 5351 2305 1559 267 3916 
13 1030 5122 2352 1579 274 4097 
14 1086 4839 2396 1596 280 4284 

 
The prediction stated that the total cases in all provinces in Java would increase, 

except Jakarta, in which there would be a declined total number of cases. The prediction 
was based on the assumption that there was no change in social interaction in the 
community. 



Spatio-temporal Modelling for Government Policy the COVID-19 Pandemic in Java 

Atiek Iriany 225 

 

CONCLUSIONS 

The model acquired in modeling the COVID-19 cases in Java was the 
GSTAR(1)(1,0,0) model. Our predicted COVID-19 case data was close to the actual number 
of COVID-19 cases in Java. A Spatio-temporal model could be used to predict the number 
of COVID-19 cases in Java. Human-to-human transmission likely had a cross-location 
impact due to an interaction between individuals. Our prediction indicates that all 
provinces in Java, but Jakarta, would likely have an increase in the total number of COVID-
19 cases for the next 14 days. 

 

ACKNOWLEDGMENTS 

We would like to thank the University of Brawijaya University for funding and support of 
this research. 

 

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