GSTAR-X-SUR Model with Neural Network Approach on Residuals


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 5(4) (2019), Pages 203-210 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: November 10, 2018 Reviewed: Pebruary 11, 2019 Accepted: May 31, 2019 
 

GSTAR-X-SUR Model with Neural Network Approach on Residuals 

Diana Rosyida1, Atiek Iriany2, Nurjannah3 

1,2,3Department of Statistics, Faculty of Mathematics and Natural Science, University of 
Brawijaya, Malang, Indonesia 

 

Email: dianarosyida94@gmail.com, atiekiriany@yahoo.com, nj_anna@ub.ac.id 

ABSTRACT  

One of the models that combine time and inter-location elements is Generalized Space Time 
Autoregressive (GSTAR) model. GSTAR model involving exogenous variables is GSTAR-X model. 
The exogenous variables which are used in GSTAR-X model can be both metrical and non-metrical 
data. The case study of this research is the forecasting of precipitation, which is the exogenous 
variable is non-metrical data of precipitation intensity of a certain location. The approach of 
parameter estimation method employed by Seemingly Unrelated Regression (SUR) model, which 
can solve the correlation between residual models.  Now, the phenomenon of precipitation 
possesses patterns and characteristics is difficult to identify, and can be interpreted as a non-linear 
model. The non-linear model is much developed by Neural Network (NN). This research employed 
GSTARX-SUR modelling with neural network approach on residuals. The data used in this research 
were the records of 10-day precipitations in four regions in West Java, namely Cisondari, Lembang, 
Cianjur, and Gunung Mas, from 2005 to 2015. The GSTARX-SUR NN modelling resulted in 
precipitation deviation average of the forecast and the actual data at 4.1385 mm. This means that 
this model can be used as an alternative in forecasting precipitation.  

Keywords: GSTAR-X, SUR, neural network, precipitation 

INTRODUCTION  

Multivariate time series analysis has been implemented in forecasting since there 
are many problems that cannot be solved with univariate forecasting [1]. One of them is 
Space Time Autoregressive (STAR) model that was first introduced by Pfeifer and Deutsch 
[2]. That model considers the element of time and location. The STAR model was 
developed by Ruchjana [3] to be a more general model for all locations, which is known 
as the Generalized Space Time Autoregressive (GSTAR). 

In time series model, there is a model that contains exogenous variables, namely 
ARIMAX and VARIMAX. Similarly, space time model has GSTARX model. Exogenous 
variable which is used in GSTAR model can be in a form of metrical data and non-metrical 
data. Some researches on GSTAR using exogenous variables with metrical data are Kurnia, 
et al [4] and Astuti, et al [5], while that of exogenous variable with non-metrical data are 
Ditago, et al [6] and Suhartono, et al [7]. 

One of the methods in estimating the parameter in multivariate analysis is 
Seemingly Unrelated Regression (SUR) [8] [9]. Parameter estimation of GSTAR model 
with Ordinary Least Square method conducted by Ruchjana [3] found residuals which are 
mostly correlated and thus the estimation value acquired is not efficient [10]. 

One of the non-linear phenomena is precipitation. Currently, precipitation 
possesses patterns which are difficult to be identified and predicted. Neural network 

mailto:dianarosyida94@gmail.com
mailto:atiekiriany@yahoo.com
mailto:nj_anna@ub.ac.id


GSTAR-X-SUR Model with Neural Network Approach on Residuals 

Diana Rosyida 204 

model is a non-linear model which has been commonly developed. Some researches 
which implement neural network in space time model are Suhartono [11] and Sulistyono, 
et al [12]. These researches employed GSTARX-SUR-NN modelling aiming at acquiring 
accurate forecast.    
  

METHODS  

The data used in this research include 10-day precipitation data in four locations in 
West Java, namely Cisondari, Lembang, Cianjur, and Gunung Mas regions. The period of 
precipitation data used is from 2005 to 2015, resulting in 360 observations from each 
observed region.  

The exogenous variable used is dummy variable. There are four dummy variables 
employed, namely precipitation of more than 20 mm in Cisondari, precipitation of more 
than 20 mm in Lembang, precipitation of more than 25 mm in Cianjur, and precipitation 
of more than 30 mm in Gunung Mas. The code of dummy variables is 1 if  the data fill the 
criteria of  four exogenous variable and 0 when the data do not. This ways is analog with 
the reasearch of Ditago [6] uses Ramadhan month criteria, in which the data in Ramadhan 
month are given 1 and 0 for the data in other months. 

Location value used in the GSRARX modelling is the normalization value of cross 
correlation initially introduced by Suhartono and Atok [13]. The normalization value of 

cross correlation taking place in the corresponding lag [14] 𝑊𝑖𝑗 =
𝑟𝑖𝑗(𝑠)

∑ |𝑟𝑖𝑘(𝑠)|𝑘≠𝑠
 where 𝑖 ≠ 𝑗 

and this value meets ∑ 𝑊𝑖𝑗 = 1𝑖≠𝑗 . 

SUR model possesses the assumption 𝐸[𝑒|𝑿𝟏, 𝑿𝟐, … , 𝑿𝒎] = 0 and 
𝐸[𝜺′𝜺|𝑿𝟏, 𝑿𝟐, … , 𝑿𝒎] = 𝛀 where 𝛀 is the variance covariance matrix [15]. 

𝛀 = [

𝜎11𝑰𝒎𝑻 𝜎12𝑰𝒎𝑻 … 𝜎1𝑚𝑰𝒎𝑻
𝜎21𝑰𝒎𝑻 𝜎22𝑰𝒎𝑻 … 𝜎2𝑚𝑰𝒎𝑻

⋮ ⋮ ⋱ ⋮
𝜎𝑚1𝑰𝒎𝑻 𝜎𝑚2𝑰𝒎𝑻 … 𝜎𝑚𝑚𝑰𝒎𝑻

] (1) 

 
where 𝑰𝒎𝑻  as the identity matrix with (𝑚𝑇 𝑥 𝑚𝑇) size and 𝜎𝑖𝑗  is the error variance from 

each equation for 𝑖 = 𝑗 and covariance error between equations 𝑖 ≠ 𝑗. Therefore, 
estimation of parameter using SUR is shown in the equation below (2) 

�̂� = (𝑿′𝛀−𝟏𝑿)−𝟏𝑿′𝛀−𝟏𝒀 (2) 

   
Essential components in the neural network model are the layer input, hidden layer, 

and output layer. Input layer used in this case is the residual of GSTAR-SUR model. This 
current research merely employs one hidden layer; however, the number of neurons used 
in the hidden layer is based on the lowest RMSE value. The output layer employed 
involves four variables. Resilient Algorithm backpropagation is used in estimating the 
neural network model. This algorithm has been employed in researches by Apriliyah, et 
al  [16] and Fadil, et al [17] in estimating the sales of electricity load.   
 
  



GSTAR-X-SUR Model with Neural Network Approach on Residuals 

Diana Rosyida 205 

RESULTS AND DISCUSSION  

The illustration of precipitation data plot in Cisondari, Lembang, Cianjur, and 
Gunung Mas is presented below. 

  
 ( a ) ( b ) 

 
 ( c ) ( d ) 

Figure 1. Precipitation Data Plot 

Figure 1.a is precipitation data plot in Cisondari, Figure 1.b for Lembang, Figure 1.c 
for Cianjur, and Figure 1.d for Gunung Mas. In Figure 1, it can be seen that the precipitation 
data patterns in these four different locations are randomly horizontal. Gunung Mas 
region has shown higher level of precipitation compared to those in the other three 
regions, with the precipitation of 50 mm being the highest. The highest precipitation in 
Cisondari is 28.15 mm; while precipitations in Lembang and Cianjur respectively are at 
24.45 mm and 36.18 mm. The increase of precipitation in these four regions has happened 
in different time period and therefore it is hard to predict.  

Precipitation data plot illustrated in Figure 1 shows that the data possess big 
variance. Therefore, Box-Cox transformation must be conducted in the precipitation data 
in the four regions. If the lambda in Box-Cox transformation equals to 1, the data can be 
classified as stationary against the variance.  

Table 1. Box-Cox Transformation  
Location Transformation 

Cisondari (𝑍1𝑡 + 1)
−0.16 

Lembang [ln(𝑍4𝑡 + 1) + 1]
0.25 

Cianjur (𝑍3𝑡 + 1)
0.21 

Gunung Mas (𝑍4𝑡 + 1)
0.12 

 
Precipitation data from the transformation based on Table 1 have shown to be 

variance stationary. It is then tested to find if the data are stationary against the average 

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GSTAR-X-SUR Model with Neural Network Approach on Residuals 

Diana Rosyida 206 

value. Dickey Fuller test result has shown that the stationary variance data are stationary 
against the average value. This means that differencing is not necessary.  

GSTAR model order is determined by the MPACF and MACF schemes; while spatial 
order used is order 1.  

 
( a ) 

 
( b ) 

Figure 2. MPACF and MACF schemes 

Figure 2.a is of a MPACF scheme and Figure 2.b is of a MACF scheme. Figure 2 shows 
that MPACF scheme cuts off at lag 1 and MACF scheme possesses sinus wave pattern, so 
the model developed is autoregressive with lag 1 [18]. Therefore, in this research, the 
order of GSTARX-SUR model used is GSTARX-SUR (11). 

Neural network modelling is to be performed on the residuals from GSTARX model 
(11). There are four inputs of input layers used, namely 𝑎1,𝑡−1, 𝑎2,𝑡−1, 𝑎3,𝑡−1, 𝑎𝑛𝑑 𝑎4,𝑡−1. The 

inputs are layers used by Suhartono and Endharta [19] as one of the inputs in neural 
network modelling. There are four neurons in the output layer of some variables used in 
GSTARX-SUR (11). Hidden layer is limited to 1 layer, where neuron used is limited to 1 to 
10 neurons. The selection of the number of neurons used in hidden layer is drawn from 
the lowest RMSE value. 

Table 2. The RMSE Value in Numbers of Neurons in Hidden Layer 
The number of 

Neurons in Hidden 
Layer 

RMSE Value The number of 
neurons in Hidden 

Layer 

RMSE Value 

1 0.1435 6 0.1346 
2 0.1430 7 0.1352 
3 0.1421 8 0.1403 
4 0.1411 9 0.1327 
5 0.1405 10 0.1303 

 
Based on Table 2, 10 neurons possess the lowest RMSE values. Therefore, the neural 

network model developed is NN (4, 10, 4). The best architectural design for neural 
network based on the residual GSTARX-SUR model (11) is shown in Figure 3. 



GSTAR-X-SUR Model with Neural Network Approach on Residuals 

Diana Rosyida 207 

The mathematical equations developed from the GSTARX-SUR (11) – NN (4,10,4) 
from each location are as follows. 

1. Cisondari 
�̂�1𝑡 = 𝑍1𝑡,𝐺𝑆𝑇𝐴𝑅𝑋 + 𝑍1𝑡,𝑁𝑁  

�̂�1𝑡 = 0.7717 𝑍1,𝑡−1 + 0.0516 𝑍2,𝑡−1 + 0.0468 𝑍3,𝑡−1 + 0.0404 𝑍4,𝑡−1 − 0.1656 𝐷1
− 0.0786 𝐷2 

𝑍1𝑡,𝑁𝑁 = −0.615 − 0.0473 𝑓(ℎ1) + 0.6355 𝑓(ℎ2) − 0.2415 𝑓(ℎ3) + 0.8345 𝑓(ℎ4)

+ 0.2273 𝑓(ℎ5) − 0.5837 𝑓(ℎ6) − 0.4523 𝑓(ℎ7) − 0.2578 𝑓(ℎ8)
− 0.6448 𝑓(ℎ9) + 1.1062 𝑓(ℎ10) 

(3) 
2. Lembang 

�̂�2𝑡 = 𝑍2𝑡,𝐺𝑆𝑇𝐴𝑅𝑋 + 𝑍2𝑡,𝑁𝑁   

�̂�2𝑡 = 0.4027 𝑍2,𝑡−1 + 0.1928 𝑍1,𝑡−1 + 0.2077 𝑍3,𝑡−1 + 0.1976 𝑍4,𝑡−1 + 0.0994 𝐷1
+ 0.0796 𝐷2 

𝑍2𝑡,𝑁𝑁 = 1.1521 + 0.0785 𝑓(ℎ1) − 0.7041 𝑓(ℎ2) − 0.1386 𝑓(ℎ3) − 1.1537 𝑓(ℎ4)

− 0.4983 𝑓(ℎ5) + 0.9914 𝑓(ℎ6) + 0.0946 𝑓(ℎ7) + 0.5594 𝑓(ℎ8)
− 0.4155 𝑓(ℎ9) − 0.5196 𝑓(ℎ10) 

(4) 
3. Cianjur 

�̂�3𝑡 = 𝑍3𝑡,𝐺𝑆𝑇𝐴𝑅𝑋 + 𝑍3𝑡,𝑁𝑁   

�̂�3𝑡 = 0.4711 𝑍3,𝑡−1 + 0.2305 𝑍1,𝑡−1 + 0.2713 𝑍2,𝑡−1 + 0.2191 𝑍4,𝑡−1 + 0.1923 𝐷1
+ 0.1197 𝐷2 + 0.2225 𝐷3 

𝑍3𝑡,𝑁𝑁 = −0.9406 + 0.2818 𝑓(ℎ1) + 0.2951 𝑓(ℎ2) − 1.1130 𝑓(ℎ3) − 2.5219 𝑓(ℎ4)

− 1.2281 𝑓(ℎ5) + 1.7891 𝑓(ℎ6) + 1.0893 𝑓(ℎ7) + 1.424 𝑓(ℎ8)
+ 0.8216 𝑓(ℎ9) + 0.7601 𝑓(ℎ10) 

(5) 
4. Gunung Mas 

�̂�4𝑡 = 𝑍4𝑡,𝐺𝑆𝑇𝐴𝑅𝑋 + 𝑍4𝑡,𝑁𝑁   

�̂�4𝑡 = 0.2083𝑍4,𝑡−1 + 0.2482 𝑍1,𝑡−1 + 0.3161 𝑍2,𝑡−1 + 0.268 𝑍3,𝑡−1 + 0.1239 𝐷1
+ 0.2187 𝐷4 

𝑍4𝑡,𝑁𝑁 = 0.1742 + 0.1249 𝑓(ℎ1) − 1.1336 𝑓(ℎ2) − 0.1868 𝑓(ℎ3) − 0.84 𝑓(ℎ4)

− 0.777 𝑓(ℎ5) + 0.7616 𝑓(ℎ6) + 2.6908 𝑓(ℎ7) + 0.8405𝑓(ℎ8)
− 0.2835 𝑓(ℎ9) − 0.4847 𝑓(ℎ10) 

(6) 
𝑓(ℎ𝑖 ) is the activation function of logistic sigmoid in the hidden unit which is defined 

as follow 

𝑓(ℎ𝑖 ) =
1

1 + 𝑒−(ℎ𝑖)
  ,     𝑖 = 1,2, … ,10 

(7) 

where, 
ℎ1 = −54.5045 + 179.7641 𝑎1,𝑡−1 + 491.5302 𝑎2,𝑡−1 − 35.5407 𝑎3,𝑡−1 − 316.5188 𝑎4,𝑡−1 
ℎ2 = 0.3103 + 7.4173 𝑎1,𝑡−1 − 1.4297 𝑎2,𝑡−1 + 1.7059 𝑎3,𝑡−1 + 0.4799 𝑎4,𝑡−1 
ℎ3 = −0.7417 − 21.0108 𝑎1,𝑡−1 − 4.3276 𝑎2,𝑡−1 − 2.0921 𝑎3,𝑡−1 + 4.6632 𝑎4,𝑡−1 
ℎ4 = 0.3549 + 14.6467 𝑎1,𝑡−1 − 3.4917 𝑎2,𝑡−1 − 0.5961 𝑎3,𝑡−1 + 0.5499 𝑎4,𝑡−1 
ℎ5 = 7.1522 − 29.3058 𝑎1,𝑡−1 + 128.514 𝑎2,𝑡−1 − 183.3922 𝑎3,𝑡−1 + 3.3083 𝑎4,𝑡−1 
ℎ6 = 0.1041 + 11.7434 𝑎1,𝑡−1 − 8.4182 𝑎2,𝑡−1 − 0.516 𝑎3,𝑡−1 + 2.8594 𝑎4,𝑡−1 

ℎ7 = −0.4036 + 0.481 𝑎1,𝑡−1 − 0.5068 𝑎2,𝑡−1 + 0.1383 𝑎3,𝑡−1 + 0.4666 𝑎4,𝑡−1 
ℎ8 = 3.1722 − 10.3635 𝑎1,𝑡−1 + 64.1415 𝑎2,𝑡−1 − 84.2384 𝑎3,𝑡−1 − 2.4503 𝑎4,𝑡−1 
ℎ9 = 9.2704 − 5.0167 𝑎1,𝑡−1 + 6.2685 𝑎2,𝑡−1 − 18.4776 𝑎3,𝑡−1 + 15.4932 𝑎4,𝑡−1 



GSTAR-X-SUR Model with Neural Network Approach on Residuals 

Diana Rosyida 208 

ℎ10 = −0.3992 − 13.9493 𝑎1,𝑡−1 − 0.9574𝑎2,𝑡−1 − 2.3135 𝑎3,𝑡−1 + 1.5081 𝑎4,𝑡−1 

 

 
Figure 3. The Best Architectural Design for GSTARX-SUR Residual Model Neural 

Network (11) 
 
The already developed model is to be forecasted and compared to the actual data in 

order to find out whether or not the models are able to illustrate the real condition of 
precipitation in Cisondari, Lembang, Cianjur, and Gunung Mas. 

 
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Error: 11.983662   Steps: 86824

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1 3 5 7 9 11131517192123252729313335

actual gstar nn

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actual gstar nn



GSTAR-X-SUR Model with Neural Network Approach on Residuals 

Diana Rosyida 209 

 
 ( c ) ( d ) 

Figure 4. Precipitation Forecast Plot in 2015 

Figure 4.a is precipitation forecast result plot of Cisondari, Figure 4.b, 4.c, and 4.d 
respectively are precipitation forecast result plots of Lembang, Cianjur, and Gunung Mas. 
The precipitation forecast result using GSTARX-SUR (11) – NN (4,10,4) possesses similar 
pattern to the actual data. However, the performance of GSTARX-SUR (11) – NN (4,10,4) 
model cannot be seen in detailed from the graph. The performance of GSTARX-SUR (11) – 
NN (4,10,4) model can be seen from the RMSE and MAD values. 

Table 3. GSTARX-SUR (11) – NN (4,10,4) Performance Model 

Regions RMSE Value MAD Value 

Cisondari 5.4012 3.5767 

Lembang 4.9883 3.3882 

Cianjur 5.7111 4.2282 

Gunung Mas 7.8792 5.3608 
 
The result of precipitation forecast in Lembang has shown the lowest RMSE and 

MAD values. The average deviation of precipitation from the forecast result and the actual 
data of Lembang is only at 3.3882 mm. On the other hand, Gunung Mas has the highest 
RMSE value and precipitation average deviation from forecast result and actual data at 
5.3882 mm.  

 

CONCLUSION 

 The Model of precipitation forecasting in Cisondari, Lembang, Cianjur, and Gunung 
Mas that are formed with GSTARX-SUR (11) - NN (4,10,4). Based on the value of 
forecasting in each location and compared by the actual data,  the result of forecasting is 
good because the average of  MAD value is 4.1385 mm, so GSTAX-SUR model with a neural 
network approach on the side can be used as a good alternative to predict precipitation. 

 

ACKNOWLEDGMENTS  

Acknowledgments to BMKG, the head of Cisondari, Chincona, Cianjur and Gunung 
Mas villages so that this research can be complete and help people in the villages. 

 
  

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GSTAR-X-SUR Model with Neural Network Approach on Residuals 

Diana Rosyida 210 

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