Hybrid Model GSTAR-SUR-NN For Precipitation Data


CAUCHY - JOURNAL OF PURE AND APPLIED MATHEMATICS 
Volume 4(2) (2016), Pages 74-80 
 

Submitted: 19 January 2016 Reviewed: 1 March 2016 Accepted: 2 May 2016 
 

Hybrid Model GSTAR-SUR-NN For Precipitation Data  

Agus Dwi Sulistyono, Waego Hadi Nugroho, Rahma Fitriani, Atiek Iriany 

Department of Statistics, Faculty of Mathematics and Natural Sciences 
Brawijaya University,  Malang 

 

Email: agus.dwi.sulistyono@gmail.com, atiekiriany@yahoo.com 

ABSTRACT 

Spatio-temporal model that have been developed such as Space-Time Autoregressive (STAR) model, 
Generalized Space-Time Autoregressive (GSTAR), GSTAR-OLS and GSTAR-SUR. Besides spatio-temporal 
phenomena, in daily life, we often find nonlinear phenomena, uncommon patterns and unidentified 
characteristics of the data. One of current developed nonlinear model is a neural network. This study is 
conducted to form a hybrid model GSTAR-SUR-NN to develop spatio-temporal model that has better 
prediction. This research is conducted on ten-daily rainfall data at 2005 - 2015 for Blimbing, Singosari, 
Karangploso, Dau, and Wagir region. Based on the results of this research, indicated that the accuracy of 
GSTAR ((1), 1,2,3,12,36)-SUR model used cross-covariance weight has relatively similar to GSTAR ((1), 1,2,3 , 
12.36)-SUR-NN (25-14-5) for  Blimbing and Singosari region with 5% error level. While Karangploso, Dau, 
and Wagir, GSTAR ((1), 1,2,3,12,36)-SUR-NN (25-14-5) model has better accuracy in predicting the 
precipitation at three locations with the value of R2prediction for each location is 0.992, 0.580, and 0.474.  

Keywords: spatio temporal, precipitation, hybrid model, neural network 

INTRODUCTION  

One of the spatio-temporal model that has been developed is Space-Time Autoregressive 
(STAR) which introduced by Pfeifer and Deutsch[1]. STAR model did not fit for the data which had 
heterogeneous characteristics of locations. It was the STAR model's weaknesses and it can be 
addressed by the Generalized Space-Time Autoregressive (GSTAR) model and GSTAR-OLS that 
developed by Ruchjana[2], [3]. The latest development of spatio-temporal models is GSTAR-SUR 
developed by Iriany [4] to address for non-stationary and seasonal pattern data. 

The use of locations weights on the formation of spatio temporal models also contribute 
to the accuracy of the model. The location weights that often used are uniform weight, inverse 
distance, and normalized cross correlation weight [5], [6]. The location weight consider the 
neighborhood between locations. For data that has a high variability, it is necessary to consider 
the location weight with variability aspects of observational data, cross covariance weights. The 
use of cross covariance weights have been studied and applied by Apanasovich and Genton [7] to 
predict pollution in California and Efromovich and Smirnova [8] to process the fMRI imaging with 
wavelate approach. 

In addition to the phenomenon of spatio-temporal, in daily life we often find nonlinear 
phenomena. Time series models that have been explained are a time series model with a linear 
approach. There is many limitations in modeling with linear approach, especially is the fulfillment 
of the assumptions underlying the linear model. Linear time series modeling is not appropriate 

mailto:agus.dwi.sulistyono@gmail.com
mailto:atiekiriany@yahoo.com


Hybrid Model GSTAR-SUR-NN For Precipitation Data 

Agus Dwi Sulistyono 75 

and difficult to do on the data with nonlinear pattern. Some nonlinear time series models have 
been developed and applied by Tong [9], Priestley [10], Lee et al. [11], as well as Granger and 
Terasvirta [12]. Nonlinear time series model that most developed and applied is Artificial Neural 
Network. Therefore, this study was conducted to form a hybrid model GSTAR-SUR-NN to develop 
spatio-temporal models that have better forecasts. 

 

THEORITICAL REVIEW 

Location Weight 

The problems that arise in the modeling of space-time is the use of location weight. There 
are several methods to determine the weight location in space-time model of the application [6], 
such as :   
a. Uniform Weight 

Uniform weights can be calculated using the formula 𝑤𝑖𝑗 =
1

𝑛𝑖
 with ni is the number of 

neighborhood locations with ith locations. Weight of the location is only used for homogeneous 
characteristics or have the same distance between locations 

b. Inverse Distance Weight 
Inverse distance weight is calculated based on the actual distance between locations. 

The closer the distance between locations, the greater weight is. Thus, a location adjacent have 
greater weight. 

c. Normalized Cross-Correlation Weight 
This weight is the result of the normalization of cross-correlation between the location 

of the corresponding time lag [5]. The normalization of cross-correlation weight was first 
introduced by Suhartono and Atok [6] and more applied research by [5]. 

d. Normalized Cross-Covariance Weight 
The use of cross covariance weight had been studied and applied by Apanasovich and 

Genton [7] to predict pollution in California, as well as Efromovich and Smirnova [8]to process 
the fMRI imaging with wavelate approach.  

 

GSTAR Model 

Ruchjana [3] suggested that GSTAR was a generalization and extension of Space Time 
Autoregresssive models (STAR) by Pfeifer [1]. The main difference are on spatial dependent and 
weight matrix. GSTAR more realistic because it is in fact more prevalent models with different 
parameters for different locations [13]. GSTAR with p order and 𝜆1, 𝜆2, … 𝜆𝑝 spatial order, GSTAR 

(𝑝𝜆1, 𝜆2, … 𝜆𝑝 ) formulated as follows [14]: 

𝑍(𝑡) = 𝝁(𝑡) + [𝚽01 + 𝚽01𝑊]𝑍(𝑡 − 1) + 𝜀(𝑡)                                                 (6) 

or 

(

𝑍1(𝑡)

𝑍2(𝑡)
⋮

𝑍𝑛 (𝑡)

) = (

𝜙01 0 … 0
0 𝜙02 … 0
⋮ ⋮ ⋱ ⋮
0 0 … 𝜙0𝑛

) (

𝑍1(𝑡 − 1)

𝑍2(𝑡 − 1)
⋮

𝑍𝑛 (𝑡 − 1)

)

+ (

𝜙11 0 … 0
0 𝜙12 … 0
⋮ ⋮ ⋱ ⋮
0 0 … 𝜙1𝑛

) (

0 𝑤12 … 𝑤1𝑛
𝑤21 0 … 𝑤2𝑛

⋮ ⋮ ⋱ ⋮
𝑤𝑛1 𝑤𝑛2 … 𝑤𝑛𝑛

) (

𝑍1(𝑡 − 1)
𝑍2(𝑡 − 1)

⋮
𝑍𝑛(𝑡 − 1)

) + (

𝑒1(𝑡)
𝑒2(𝑡)

⋮
𝑒𝑛 (𝑡)

) 

    (7) 

Seemingly Unrelated Regression (SUR) 

Seemingly Unrelated Regression (SUR) is an equation that parameter estimation use 
General Least Square (GLS). Iriany (2013) explains that GLS is the regression coefficient estimator 



Hybrid Model GSTAR-SUR-NN For Precipitation Data 

Agus Dwi Sulistyono 76 

that count the relationship between the equation error. The error value is obtained from the 
estimated ordinary least squares (OLS) that will be used in the calculation to estimate the 
regression coefficients in the SUR system equation. SUR models with M equation expressed by: 

𝒚𝒊 = 𝑿𝒊 𝜷𝒊 + 𝜺𝒊,    𝑖 = 1, … , 𝑁 (8) 
where 𝒚𝒊 is vector with size R×1, 𝑋𝑖 ’s size is R×𝑘𝑖  dan 𝛽𝑖  vector with size 𝑘𝑖 × 1.  

According to Greene [15], equation (8) is SUR model with the assumption 
𝐸[𝜀|𝑿𝟏,𝑿𝟐, … , 𝑿𝑵] = 0 and 𝐸[𝜀𝜀

′|𝑿𝟏,𝑿𝟐, … , 𝑿𝑵] = 𝛀 with 𝛀 is variance-covariance matrices. 

GLS method use the error variance : 
Cov(𝜀) = 𝐸(𝜀𝜀′) = σ2𝚺 = 𝛀. 

Matrix 𝛀 describe the error correlation with : 

𝛀 = E(𝜀𝜀 ′) = [

E(e1e1
′ ) E(e1e2

′ ) … E(e1e𝑁
′ )

E(e2e1
′ ) E(e2e2

′ ) … E(e2e𝑁
′ )

⋮
E(e𝑁e1

′ )
⋮

E(e𝑁e2
′ )

⋱
…

⋮
E(e𝑁e𝑁

′ )

] 

Since 𝐸(𝜀𝑖𝜀𝑗 ) = 𝜎𝑖𝑗 𝐼𝑇, 

𝛀 = [

σ11𝐈𝑁𝑇 σ12𝐈𝑁𝑇 … σ1𝑁𝐈𝑁𝑇
σ21𝐈𝑁𝑇 σ22𝐈𝑁𝑇 … σ2𝑁 𝐈𝑁𝑇

⋮
σ𝑁1𝐈𝑁𝑇

⋮
σ𝑁2𝐈𝑁𝑇

⋱
…

⋮
σ𝑁𝑁 𝐈𝑁𝑇

] = 𝚺 ⊗ 𝐈𝑁𝑇  

𝚺 = [

σ11 σ12 … σ1𝑚
σ21 σ22 … σ2𝑚

⋮
σ𝑚1

⋮
σ𝑚2

⋱
…

⋮
σ𝑚𝑚

] (9) 

with 𝐈𝑁𝑇  is identity matrix sized (𝑁𝑇 ×  𝑁𝑇) and 𝚺 is matrix sized (𝑁 × 𝑁) with 𝜎𝑖𝑗  error variance 

from each equation for i= j and error covariance between equation for i≠ j. 
The parameter model estimation is obtained by estimating parameter 𝜷 in equation (8). 

𝜺∗
′ 𝜺∗ = 𝜺

′𝛀−𝟏𝛆

= 𝒀′𝛀−𝟏𝐘 − 𝟐𝐘′𝛀−𝟏𝐗𝜷 + 𝜷′𝑿′𝛀
−𝟏

𝐗 𝜷
(10) 

Equation (10) is differenced by 𝜷 and equal to zero. So : 

(𝑿′𝛀
−𝟏

𝑿)�̂� = 𝑿′𝛀
−𝟏

𝒀

(𝑿′𝛀−𝟏𝑿)−𝟏(𝑿′𝛀−𝟏𝑿)�̂� = (𝑿′𝛀−𝟏𝑿)−𝟏𝑿′𝒀

𝑰�̂� =  (𝑿′𝛀−𝟏𝑿)−𝟏𝑿′𝛀
−𝟏

𝒀

�̂� =  (𝑿′𝛀−𝟏𝑿)−𝟏𝑿′𝛀
−𝟏

𝒀

 

 

Hybrid Model GSTAR-SUR-NN 

Hybrid model by integrating the neural network with conventional forecasting model was 
proved capable of producing more accurate forecasts. Some hybrid models using neural network 
are Feed Forward Neural Network for time series data [16], Auto Regressive Integrated Moving 
Average With Exogenous Factor-Neural Network (ARIMAX-NN) for data Inflation in Indonesia 
[17], and Neural Network - Multiscale Autoregressive (NN-MAR) for forecasting the number of 
tourists [18]. 

General Space-Time Autoregressive With Seemingly Unrelated Regression Neural 
Network (GSTAR-SUR-NN) is an integration/fusion (hybrid) between GSTAR model with neural 
network. GSTAR-SUR is used to obtain the most suitable NN architecture, so it can obtain the best 
forecasting performance. Architecture in the meaning is the amount of input variables that will be 
used in modeling NN. 
 
 



Hybrid Model GSTAR-SUR-NN For Precipitation Data 

Agus Dwi Sulistyono 77 

RESULTS AND DISCUSSION 

Based on the result of descriptive statistics analysis, we can describe the descriptive 
statistics of precipitation data for each location : 

 
Table 1. Description of Precipitation Data in Five Location 

Location N 
Mean 
(mm) 

Standard 
Deviation (mm) 

Minimum 
(mm) 

Maximum 
(mm) 

Blimbing 360 5.682 6.909 0 33.5 

Singosari 360 3.93 5.575 0 41.75 

Karangploso 360 4.302 5.71 0 25.36 

Dau 360 4.564 5.825 0 36.38 

Wagir 360 7.08 8.187 0 43.63 

 

Table 1 shows that precipitation of each location has a high variability. It can be seen that 
the standard deviation of precipitation of all location greater than the average. High variability 
indicates that there is fluctuation to the extreme point of precipitation, especially during the rainy 
season. Here is the result of homogeneity test of variance the precipitation data: 

 

 

Figure 1. The Result of Homogeneity Test of Variance 

 

Figure 2 describe the results of homogeneity test of variance precipitation data using Bartlett and Levene 
test that was obtained p-value of 0.000 (p < 0.05), indicating that the precipitation data at Blimbing, 
Karangploso, Singosari, Wagir, and Dau region have different variance. Model identification is done by 
comparing the value of AIC at some time lag. Here the value of AIC with VARMAX procedure in SAS: 

Table 2. The Value of AIC with VARMAX Procedure 

Time Lag AIC Value 

1 14.67906 

2 14.52868 

3 14.52445 

 



Hybrid Model GSTAR-SUR-NN For Precipitation Data 

Agus Dwi Sulistyono 78 

Based on table 2 above, indicated that at the time lag of 1-3, the lowest AIC is in 3rd time 
lag. Thus, the order GSTAR model is GSTAR ((1), 1,2,3)-SUR. Based on ACF of precipitation data at 
each location there is seasonal pattern in the time lag of 12 and 36. Therefore, the appropriate 
model is GSTAR ((1) (1,2,3,12 , 36)) - SUR. The architecture of GSTAR-SUR-NN model is as follows: 

 

Figure 2. The architecture of GSTAR((1),1,2,3,12,36)-SUR-NN (25-14-5) Model 

 

The neural network architecture  consist of 25 input variables, 14 hidden layers, dan 5 
output layers. Therefore, hybrid model of GSTAR-SUR-NN is GSTAR((1),1,2,3,12,36)-SUR-NN (25-
14-5). Follow is the result of validation test of GSTAR ((1)(1,2,3,12,36))-SUR model and 
GSTAR((1),1,2,3,12,36)-SUR-NN (25-14-5) model : 

 
Table 3. The Result of GSTAR-SUR and GSTAR-SUR-NN Model Validation Test 

Model t-statistics p-value 

GSTAR((1),1,2,3,12,36)-SUR 2.619 0.010 

GSTAR((1),1,2,3,12,36)-SUR-NN (25-14-5) 0.332 0.741 

 

Based on the results of validation test of GSTAR ((1), 1,2,3,12,36)-SUR using cross 
covariance weight in Table 3, showed that, at α = 5%, GSTAR ((1),1,2,3,12,36)-SUR model has p-
value less than 0.05 which implies that there is a significant difference between average of actual 
precipitation data with average of predicted results. Or in other words, GSTAR ((1), 1,2,3,12,36)-
SUR using cross-covariance weights still have low accuracy. While, GSTAR ((1), 1,2,3,12,36) -SUR-
NN (25-14-5) was obtained p-value of 0.741. P-value is more than 0.05 implies that there is no 
significant difference between the average of actual precipitation with the average precipitation 
predicted results. Or in other words, GSTAR ((1),1,2,3,12,36)-SUR-NN (25-14-5) model have high 
accuracy. From this comparison, it has been proven that GSTAR ((1), 1,2,3,12,36)-SUR-NN (25-14-
5) model has better accuracy rate in predicting the precipitation data. 

 
Table 4. The Comparison GSTAR-SUR and GSTAR-SUR-NN Model Performance 

Location 

GSTAR-SUR Model GSTAR-SUR-NN Model 

Data Testing 
RMSE R2prediction 

Data Testing 
RMSE R2prediction 

Blimbing 
10.132 

0.539 
10.672 

0.640 

Singosari 0.451 0.458 



Hybrid Model GSTAR-SUR-NN For Precipitation Data 

Agus Dwi Sulistyono 79 

Karangploso 0.623 0.992 

Dau 0.527 0.580 

Wagir 0.365 0.474 

 

Table 4 explains the performance comparison between GSTAR ((1),1,2,3,12,36)-SUR with 
the hybrid model GSTAR ((1), 1,2,3,12,36)-SUR-NN (25-14 -5). From these comparison, indicated 
that GSTAR-SUR model has RMSE value lower than the hybrid model GSTAR-SUR-NN. Judging 
from the value of RMSE, GSTAR-SUR model have better performance because it produces lower 
RMSE value. But, if it judge from the R2prediction value, indicated that the R2prediction value on 
the hybrid model GSTAR-SUR-NN is higher than GSTAR-SUR model. In fact, the R2prediction value 
of Karangploso region reached 0.992. Based on this comparison, it is proved that the hybrid model 
GSTAR ((1),1,2,3,12,36)-SUR-NN (25-14-5) has better accuracy rate in predicting precipitation in 
Malang especially Blimbing, Singosari, Karangploso, Dau, and Wagir region. 

 

CONCLUSION 

Based on the results of this study concluded that the hybrid model GSTAR ((1), 
1,2,3,12,36)-SUR-NN (25-14-5) produces precipitation forecast better than GSTAR 
((1),1,2,3,12,36)-SUR model. 

 

REFERENCE 

[1] P. E. Pfeifer and S. J. Deutsch, “Identification and Interpretation of First Order Space-Time ARMA 
Models,” Technometrics, vol. 22, no. 3, pp. 397–408, 1980. 

[2] B. N. Ruchjana, “Study on the Weight Matrix in the Space-Time Autoregressive Model,” in Proceeding 
of the 10th International Symposium on Applied Stochastic Models and Data Analysis (ASMDA), France, 
2001, vol. 2, pp. 789–794. 

[3] B. N. Ruchjana, “A Generalized Space Time Autoregressive Model and its Application to Oil Production 
Data,” Institut Teknologi Bandung, Bandung, 2002. 

[4] A. Iriany, Suhariningsih, B. N. Ruchjana, and Setiawan, “Prediction of Precipitation Data at Batu Town 
Using the GSTAR (1,p)-SUR Model,” J. Basic Appl. Sci. Res., vol. 3, no. 6, pp. 860–865, 2013. 

[5] S. Suhartono and S. Subanar, “The Optimal Determination Of Space Weight in Gstar Model by Using 
Cross-Correlation Inference,” Quant. METHODS, vol. 2, no. 2, pp. 45–53, Dec. 2006. 

[6] Suhartono and R. M. Atok, “Pemilihan Bobot Lokasi yang Optimal pada Model GSTAR,” in Prosiding 
Konferensi Nasional Matematika XIII, Semarang, 2006. 

[7] T. V. Apanasovich and M. G. Genton, “Cross-covariance functions for multivariate random fields based 
on latent dimensions,” Biometrika, vol. 97, no. 1, pp. 15–30, Mar. 2010. 

[8] E. Smirnova, “Statistical Analysis of Large Cross-Covariance and Cross-Correlation Matrices Produced 
by fMRI Images,” J. Biom. Biostat., vol. 5, no. 2, 2013. 

[9] H. Tong, Non-Linear Time Series: A Dynamical System Approach. Oxford University Press, 1993. 
[10] M. B. Priestley, Non-Linear and Non-Stationary Time Series Analysis, 2nd edition, 2nd ed. London: 

Academic Press, 1991. 
[11] T.-H. Lee, H. White, and C. W. J. Granger, “Testing for neglected nonlinearity in time series models: A 

comparison of neural network methods and alternative tests,” J. Econom., vol. 56, no. 3, pp. 269–290, 
Apr. 1993. 

[12] C. W. . Granger and T. Terasvirta, Modeling   Nonlinear   Economic Relationships. United Kingdom: 
Oxford University Press. 

[13] W. Dhoriva Urwatul and S. Suhartono, “PERAMALAN DERET WAKTU MULTIVARIAT SEASONAL PADA 
DATA PARIWISATA DENGAN MODEL VAR-GSTAR,” in Seminar Nasional Matematika dan Pendidikan 
Matematika 2009, 2009. 

[14] S. A. Borovkova, H. P. Lopuha, and B. N. Ruchjana, “Generalized S-TAR with Random Weights,” in the 
17th International Workshop on Statistical Modeling, Chania-Greece. 

[15] W. H. Greene, Econometric analysis, 7th ed. Boston: Prentice Hall, 2012. 
[16] Suhartono, “Feedforward Neural Networks untuk Pemodelan Runtun Waktu,” Universitas Gajah 

Mada, Yogyakarta, 2007. 



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Agus Dwi Sulistyono 80 

[17] H. Suryono, “Auto Regressive Integrated Moving Average With Exogeneous Factor-Neural Network 
(ARIMAX-NN) pada Data Inflasi di Indonesia,” Institut Teknologi Sepuluh November, Surabaya, 2009. 

[18] B. Sutijo, S. Suhartono, and A. J Endharta, “Forecasting Tourism Data Using Neural Networks-
Multiscale Autoregressive Model,” J. Mat. Sains, vol. 16, no. 1, pp. 35–42, 2011. 

 
 
 


	ABSTRACT
	INTRODUCTION
	THEORITICAL REVIEW
	Location Weight
	GSTAR Model
	Seemingly Unrelated Regression (SUR)
	Hybrid Model GSTAR-SUR-NN

	RESULTS AND DISCUSSION
	CONCLUSION
	REFERENCE