CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 5(1)(2017), Pages 8-14 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 
 

Submitted: 26 July 2016 Reviewed: 18 October 2017 Accepted: 2 November 2017 
DOI: http://dx.doi.org/10.18860/ca.v5i1.4305 

Estimation of Geographically Weighted Regression Case Study on Wet Land Paddy 
Productivities in Tulungagung Regency 

Danang Ariyanto, Henny Pramoedyo, Suci Astutik 

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

 

Email: danangariyanto75@gmail.com, pramoedyohp@yahoo.com, suci_sp@yahoo.com  
 

ABSTRACT 

Regression is a method connected independent variable and dependent variable with estimation parameter 
as an output. Principal problem in this method is its application in spatial data. Geographically Weighted 
Regression (GWR) method used to solve the problem. GWR is a regression technique that extends the 
traditional regression framework by allowing the estimation of local rather than global parameters. In other 
words, GWR runs a regression for each location, instead of a sole regression for the entire study area. The 
purpose of this research is to analyze the factors influencing wet land paddy productivities in Tulungagung 
Regency. The methods used in this research is GWR using cross validation bandwidth and weighted by 
adaptive Gaussian kernel function. This research using four variables which are presumed affecting the wet 
land paddy productivities such as:  the rate of rainfall(X1), the average cost of fertilizer per hectare(X2), the 
average cost of pesticides per hectare(X3) and Allocation of subsidized NPK fertilizer of food crops sub-
sector(X4). Based on the result, X1, X2, X3 and X4T has a different effect on each District. So, to improve the 
productivity of wet land paddy in Tulungagung Regency required a special policy based on the GWR model 
in each district. 
 
Keywords: spatial data, geographically weighted regression, GWR, cross validation, adaptive gaussian 
kernel 

INTRODUCTION  

The conventional spatial analysis techniques use a single equation to assess the overall 
relationships between the dependent and independent variables across space, known as a global 
analytic approach. One important assumption underlying this approach is that the relationships 
of interest are stationary or homogeneous spatially. While the global perspective is effective in 
handling spatial dependence and generating less unbiased estimates (than the non-spatial 
modeling), it is not capable of exploring spatial non-stationarity (or heterogeneity) or identifying 
place-specific associations [1].  
 Geographically weighted regression (GWR) is a local spatial statistical technique used 
to analyze spatial non-stationarity, defined as when the measurement of relationships among 
variables differs from location to location. Unlike conventional regression, which produces a 
single regression equation to summarize global relationships among the explanatory and 
dependent variables, GWR generates spatial data that express the spatial variation in the 

mailto:danangariyanto75@gmail.com
mailto:pramoedyohp@yahoo.com
mailto:suci_sp@yahoo.com


Estimation of Geographically Weighted Regression Case Study on Wet Land Paddy Productivities in 
Tulungagung Regency 

 

Danang Ariyanto 9 

relationships among variables [2]. This approach includes locational information and smoothing 
techniques into regression models. In contrast to the global approach, GWR has proved to be a 
useful local spatial analysis tool that helps researchers to generate nuanced insights into existing 
literature [1]. In this research, the parameter estimation on the GWR method requires a weights 
matrix that calculated by adaptive gaussian kernel function. The data weighting is according to 
the proximity of the 𝑖-th observation location. Cross validation is used for estimating the kernel 
bandwidth.  The case study investigated the productivity of wet land paddy in Tulungagung 
Regency, there are 18 districts in Tulungagung Regency and 1 district has no wet land paddy 
productivity. This research using four variables which are presumed affecting the wet land paddy 
productivities such as:  the rate of rainfall(X1), the average cost of fertilizer per hectare(X2), the 
average cost of pesticides per hectare(X3) and Allocation of subsidized NPK fertilizer of food crops 
sub-sector(X4). The final result of this GWR method will be obtained productivity model of wet 
land paddy in Tulungagung regency. In addition, the mapping of paddy productivity per district is 
expected to be useful and add information especially in agriculture to increase wet land paddy 
productivity in Tulungagung Regency. 

 

METHODS 

Geographically weighted regression is an extension of the traditional multiple linear 
regression toward a local regression, in which regression coefficients are specific to a location 
rather than being global estimates. the specification of a basic GWR model is: 

𝑦𝑖 = 𝛽0(𝑢𝑖, 𝑣𝑖 ) + ∑ 𝛽𝑘 (𝑢𝑖 , 𝑣𝑖 )𝑥𝑖𝑘 + 𝜀𝑖  ;    𝑖 = 1,2, … , 𝑛                                                             (1)

𝑝

𝑘=1

 

where yi is the dependent variable at location i, 𝑥𝑖𝑘  is the value of the kth explanatory variable at 

location i, the 𝛽𝑘 (𝑢𝑖 , 𝑣𝑖 )𝑥𝑖𝑘  is the local regression coefficient for the kth explanatory variable at 
location i, 𝛽0(𝑢𝑖 , 𝑣𝑖 ) is the intercept parameter at location i, and 𝜀𝑖  is the random disturbance at 
location i, which may follow an independent normal distribution with zero mean and 
homogeneous variance [3].  

To facilitate the exposition, it is convenient to express the GWR model in matrix notation: 

     𝒀𝒘 = 𝑿𝒘𝜷𝒘 + 𝜺𝒘                                                                          (2) 

Weighted Least Square (WLS) is used for Geographically Weighted Regression parameter 
estimation, so: 

𝐿  = 𝜺𝒘
𝑻𝜺𝒘 

      =  (𝒀𝒘 − 𝑿𝒘𝜷𝒘)
𝑻(𝒀𝒘 − 𝑿𝒘𝜷𝒘) 

      =  𝒀𝒘
𝑻𝒀𝒘 −  𝜷𝒘

𝑻
𝑿𝒘

𝑻𝒀𝒘 − 𝒀𝒘
𝑻𝑿𝒘 𝜷𝒘 +  𝜷𝒘

𝑻
𝑿𝒘

𝑻𝑿𝒘𝜷𝒘   

      = 𝒀𝒘
𝑻𝒀𝒘 − 𝟐𝜷𝒘

𝑻
𝑿𝒘

𝑻𝒀𝒘 + 𝜷𝒘
𝑻

𝑿𝒘
𝑻𝑿𝒘𝜷𝒘                                                                    (3) 

As we learned in calculus, a univariate optimization involves taking the derivative and setting 
equal to 0. This gives us, 

𝜕𝐿

𝜕𝜷𝒘
=

𝜕(𝒀𝒘
𝑻𝒀𝒘 − 𝟐𝜷𝒘

𝑻
𝑿𝒘

𝑻𝒀𝒘 + 𝜷𝒘
𝑻

𝑿𝒘
𝑻𝑿𝒘 𝜷𝒘)

𝜕𝜷𝒘
= 0 

𝜕𝐿

𝜕𝜷𝒘
= −𝟐𝑿𝒘

𝑻𝒀𝒘 + 𝟐𝑿𝒘
𝑻𝑿𝒘�̂�𝒘 = 0 

𝜕𝐿

𝜕𝜷𝒘
= −𝑿𝒘

𝑻𝒀𝒘 + 𝑿𝒘
𝑻𝑿𝒘�̂�𝒘 = 0      



Estimation of Geographically Weighted Regression Case Study on Wet Land Paddy Productivities in 
Tulungagung Regency 

 

Danang Ariyanto 10 

𝑿𝒘
𝑻𝑿𝒘�̂�𝒘 = 𝑿𝒘

𝑻𝒀𝒘        

�̂�𝒘   = (𝑿𝒘
𝑻𝑿𝒘)

−𝟏𝑿𝒘
𝑻𝒀𝒘                                                                                                         (4) 

If     𝑾
1

2𝒀 = 𝑾
1

2𝑿𝜷 + 𝑾
1

2𝜀   is equal to 𝒀𝒘 = 𝑿𝒘 𝜷𝒘 + 𝜺𝒘   so, 

=  [(𝑾
1

2𝑿)𝑻𝑾
1

2𝑿]
−𝟏

(𝑾
1

2𝑿)𝑻𝑾
1

2𝒀 

= (𝑿𝑻 𝑾
1

2𝑾
1

2𝑿)−𝟏𝑿𝑻𝑾
1

2𝑾
1

2𝒀 

             �̂�𝒘 = (𝑿
𝑻 𝑾𝑿)−𝟏𝑿𝑻𝑾𝒀                                       (5)

      

For each i-th point, the Geographically Weighted Regression model parameter estimation is 
performed by matrix operation: 
 

�̂�(𝑢𝑖 , 𝑣𝑖 )  = (𝑿
𝑻 𝑾(𝑢𝑖 , 𝑣𝑖 ) 𝑿)

−𝟏𝑿𝑻 𝑾(𝑢𝑖 , 𝑣𝑖 ) 𝒀                𝑖 = 1,2, … , 𝑛                                                      (7) 

with, 

𝑾𝒊𝒋(𝑢𝑖 , 𝑣𝑖 ) = [

𝒘𝑖1 0 ⋯ 0
0 𝒘𝑖2 ⋯ 0
⋮ ⋮ ⋱ ⋮
0 0 ⋯ 𝒘𝑖𝑛

] 

The first step to estimate parameters in GWR, it is important to decide the spatial 
weighting matrix, which can be calculated by different methods. One method is to specify Wij as a 
continuous and monotonic decreasing function of distance dij between points i and j. For adaptive 
kernel size, the weight of each point can be calculated by applying the Gaussian function [3]: 

     𝑤𝑖𝑗 (𝑢𝑖, 𝑣𝑖 ) = 𝑒𝑥𝑝(− (
𝑑𝑖𝑗

ℎ𝑖(𝑞)
)

2

)                                                     (8) 

where 𝑤𝑖𝑗 (𝑢𝑖, 𝑣𝑖 ) is the weight of location j in the space at which data are observed for estimating 

the dependent variable at location i, andℎ𝑖(𝑞) is referred as a bandwidth. 𝑑𝑖𝑗  is eucledian distance 

between points i and j, 𝑑𝑖𝑗 = √(𝑢𝑖 − 𝑢𝑗 )
2 + (𝑣𝑖 − 𝑣𝑗 )

2 . Bandwidth is used to specifies how the extent 

of the kernel should be determined. It controls the degree of smoothing in the model. Cross-
validation (CV) is an iterative process that searches for the kernel bandwidth that minimizes the 
prediction error of all the y(s) using a subset of the data for prediction [1]. 

     𝐶𝑉(ℎ) = ∑ (𝑦𝑖 − �̂�≠𝑖 (ℎ))
2𝑛

𝑖=1                                         (9) 
where �̂�≠𝑖 (ℎ) is the predicted value of observation 𝑖 with calibration location 𝑖 left out of the 
estimation dataset. 

 The second step is testing for spatial heterogeneity. Spatial heterogeneity indicates the 
variation between location. So, each location has different relationship structures and parameters. 
Spatial data heterogeneity can be tested using the Breach-Pagan (BP) test [4]: 

H0: 
1 1 2 2

2 2 2 2

( , ) ( , ) ( , )n nu v u v u v
       ,  

2 2

( , )i iu v
   (there is no spatial heterogeneity)  

H1:    at least one  
2 2

( , )i iu v
   (there is spatial heterogeneity) 

   𝐵𝑃 =  (
1

2
) 𝒇 𝑇 𝒁 (𝒁𝑇 𝒁)−1𝒁𝑇 𝒇 +  (

𝟏

𝑇
) [

𝑒𝑇𝑾𝑒

𝝈𝟐
  ]

𝟐

~ 𝜒2(𝑘+1)               (10) 



Estimation of Geographically Weighted Regression Case Study on Wet Land Paddy Productivities in 
Tulungagung Regency 

 

Danang Ariyanto 11 

Where  𝒇𝒊 =
𝒆𝒊

𝝈𝟐
− 𝟏, 𝒆𝒊 is a vector of OLS residuals,  𝝈

𝟐 is the variance based on OLS residual,  

T= Trace [ WTW + W2] and  𝒁 is an N by (k+1) matrix of normal standard score (z). 

 The third step is testing parameters of GWR model partially using t test with hypothesis 
as follows [5]: 

H0: 𝛽𝑗 (𝑢𝑖 , 𝑣𝑖 ) = 0 

H1: 𝛽𝑗 (𝑢𝑖 , 𝑣𝑖 ) ≠ 0 ;  𝑗 = 1,2, … , 𝑘 

t test statistic can be written as follows: 

    
�̂�𝑘(𝑢𝑖,𝑣𝑖)

�̂�√𝑐𝑗𝑗
 ∼   𝑡(𝑛−𝑘−1)   (11) 

Where 𝑐𝑗𝑗  is a diagonal element of the CCT matrix, with 𝑪 =  (𝑿
𝑻 𝑾(𝒖𝒊, 𝒗𝒊)𝑿)

−𝟏𝑿𝑻𝑾(𝒖𝒊, 𝒗𝒊). 

 The fourth step is testing parameter of GWR model simultaneously, the hypothesis is: 

H0: ( , )
j i i j

u v  , where 𝑗 = 1, 2, … 𝑘. 

H1: at least one ( , )
j i i

u v has a relation with location ( , )i iu v  

The statistic test is: 

2
1

2

2

1
1

2 *

( , 1)
0

( ) /

( ) / 1 n k

SSE H

F
SSE H n k 







 

 
 
 

 
   (12) 

where: 

0
( )SSE H 

T T
y (I - L) (I - L)y  

1
( ) SSE H 

T
y (I - S)y  

  1 trace 
T

I - L (I - L)  

  
2

2
trace 

T
I - L (I - L)  

1

1 1 1 1 1

1

2 2 2 2 2
( )

1

( , ) ( , )

( , ) ( , )

( , ) ( , )

T T T

T T T

n n

T T T

n n n n n

x X W u v X X W u v

x X W u v X X W u v

x X W u v X X W u v









  
  

 
 

  
 
 
  

  

L  

  
T -1 T

S X(X X) X  

I    = identity matrix ordo n 
 

RESULTS AND DISCUSSION 

The results of Breusch-Pagan (BP) test in wet land paddy productivities in Tulungagung 
Regency is: 

𝐵𝑃 =  (
1

2
) 𝒇 𝑇 𝒁 (𝒁𝑇𝒁)−1𝒁𝑇 𝒇 +  (

𝟏

𝑇
) [

𝑒𝑇 𝑾𝑒

𝝈𝟐
  ]

𝟐

=  7438,636 



Estimation of Geographically Weighted Regression Case Study on Wet Land Paddy Productivities in 
Tulungagung Regency 

 

Danang Ariyanto 12 

Because BP test >  𝜒2(0,05; 5) =  11,070, the decision is Reject 𝐻0. There is spatial 

heterogeneity in wet land paddy productivities in Tulungagung Regency, so we can use 
geographically weighted regression to estimate parameter model wet land paddy productivities 
in Tulungagung Regency. 

The next step is to determine the parameter estimation of the GWR model based on the 
equation 7. Table 1 show the parameters estimation GWR model. 

 
Table 1. Parameters estimation GWR model  

No district       
0

     
1
  

2
  

3
  

4
  

1 Besuki 26,553 -0,02175 0,0000165 0,0000363 -0,0015 

2 Bandung 40,055 0,01395 0,0000014 0,0000374 -0,00065 

3 Pakel 27,372 -0,02315 0,0000179 0,0000311 -0,0016 

4 Campurdarat 35,107 -0,01342 0,0000077 0,0000396 -0,00154 

5 Kalidawir 42,804 0,01075 -0,0000009 0,0000370 -0,00036 

6 Pucanglaban 59,237 0,12682 -0,0000260 0,0000264 0,007461 

7 Rejotangan 55,276 0,08255 -0,0000189 0,0000297 0,005735 

8 Ngunut 40,515 0,03892 -0,0000185 0,0000672 0,010068 

9 Sumbergempol 41,269 0,05628 -0,0000239 0,0000751 0,011815 

10 Boyolangu 40,946 0,06148 -0,0000224 0,0000706 0,0117 

11 Tulungagung 41,456 0,05585 -0,0000207 0,0000664 0,011093 

12 Kedungwaru 41,122 0,05185 -0,0000186 0,0000621 0,01075 

13 Ngantru 42,624 0,03504 -0,0000128 0,0000508 0,007392 

14 Karangrejo 45,889 0,04003 -0,0000111 0,0000406 0,005076 

15 Kauman 59,895 0,10582 -0,0000251 0,0000278 0,0076 

16 Gondang 59,386 0,17687 -0,0000308 0,0000234 0,008816 

17 Pagerwojo 60,541 0,16031 -0,0000305 0,0000244 0,009065 

18 Sendang 41,348 0,02341 -0,0000006 0,0000365 -0,000099 

 

The GWR model for district Besuki based on Table 1 can be written as follows: 

  �̂�1 = 26,553 - 0,0217x1 + 0,0000165x2 + 0,0000363x3 - 0,00150x4     (13) 

 Other GWR models for the other districts has the same way as in equation 8 based on 
Table 1. Testing parameters of GWR model simultaneously conducted to determine the effect of 
weighting in the process of parameter estimation on the case of paddy productivity in 
Tulungagung Regency. Results of simultan parameter test based on equation 12. F test is 8,904. 
Value of F test is greater than F (0,05;10,3) = 8,785. This shows that simultaneously the predictor 
variables X1, X2, X3 and X4 have significant effect spatially on response variable.  The GWR model 
of each district is formed on the basis of influential parameters, so partial parameter testing is 
performed by Table 2 using t test. 
 
 
 
 
 
 



Estimation of Geographically Weighted Regression Case Study on Wet Land Paddy Productivities in 
Tulungagung Regency 

 

Danang Ariyanto 13 

Table 2. t test statistics for parameter of GWR model 

No District t test 
statistic 

0
  

t test 
statistic 

1
  

t test 
statistic 

2
  

t test 
statistic 

3
  

t test 
statistic 

4
  

1 Besuki 5,429 -1,23768 2,974011 7,616609 -1,81654 

2 Bandung 12,447 0,907966 0,456556 11,73849 -0,89729 

3 Pakel 5,814 -1,27889 2,910577 4,486399 -1,91835 

4 Campurdarat 10,604 -0,82625 2,479856 11,72245 -1,91886 

5 Kalidawir 18,321 0,886951 -0,41056 13,04989 -0,58983 

6 Pucanglaban 13,264 5,188448 -5,04069 7,326105 4,716729 

7 Rejotangan 14,778 3,647647 -4,34952 8,891485 3,844718 

8 Ngunut 19,811 2,362104 -8,07605 15,72467 11,62082 

9 Sumbergempol 14,223 3,046957 -5,58308 10,28998 10,56913 

10 Boyolangu 14,431 3,073174 -6,91394 12,4301 11,11775 

11 Tulungagung 19,018 3,071231 -7,73936 13,62258 11,04025 

12 Kedungwaru 19,701 2,941217 -8,31189 14,96989 11,01064 

13 Ngantru 18,828 2,111468 -7,95199 18,32336 9,755659 

14 Karangrejo 17,363 2,105942 -6,30701 13,46622 5,196659 

15 Kauman 13,516 3,402 -4,47746 6,19037 3,55978 

16 Gondang 5,404 1,296335 -4,49924 2,7131 3,188156 

17 Pagerwojo 9,567 2,655057 -4,6735 5,152779 3,895126 

18 Sendang 12,836 1,525182 -0,21392 11,48837 -0,13862 

                    Significant if t test > t(0,05;13) = 2,16 

 

 Based on Table 2, The Farmers need to add the average cost of pesticides per hectare(X3) 
to each district because the effect is positive significant to the wet land paddy productivity. In the 
districts of Ngantru, Karangrejo and Gondang need to add the average cost of fertilizer per 
hectare(X2) because it has a significant positive effect on paddy productivity. The addition of X2 
and X3 is not continuously. There are 12 districts that need to be added by the allocation of 
subsidized NPK fertilizer of food crops sub-sector(X4) because it proved to have a significant 
positive effect on wet land paddy productivity. The district is grouping according to the variables 
that significantly affect the GWR model, the result is listed at Tabel 3.  
 

Table 3. The Grouping according to the significant variables of GWR model 

district Group Significant variables 

Pucanglaban, Rejotangan, Ngunut, 
Sumbergempol, Boyolangu, 

Tulungagung, Kedungwaru, Kauman, 
Pagerwojo 

Yellow X1, X2, X3 and X4 

Ngantru, Karangrejo, Gondang Red X2, X3 and X4 

Besuki, Pakel, Campurdarat Purple X2 and X3 

Bandung, Kalidawir, Sendang Green X3 

 
 Based on Table 3, a spread pattern of variables that has a significant effect on paddy 
productivity by districts in Tulungagung Regency presented in Figure 1. 



Estimation of Geographically Weighted Regression Case Study on Wet Land Paddy Productivities in 
Tulungagung Regency 

 

Danang Ariyanto 14 

 
Figure 1. The spread pattern of variables that significantly affected by district on wet land paddy 

productivities in Tulungagung regency 
 

The yellow group is the group where the four variables, X1, X2, X3 dan X4 is significant to 
the productivity of wet land paddy. The yellow group consisted of 9 districts. Only Variable X2, X3 
and X4 had a significant effect on the red group, there were 3 districts in the red group. Purple 
group is a group with 2 variables that have significant effect that is X2 and X3. There are 3 districts 
that enter the purple group. The last group is green group where only X3 has significant effect. 
Green group consists of 3 districts. The District of Tanggunggunung has no wet land paddy 
productivity so the colour is white. 

CONCLUSION 

 Based on the result, X1, X2, X3 and X4 has a different effect on each district. So, to improve 
the productivity of wet land paddy in Tulungagung Regency required a special policy based on the 
GWR model in each district. 
 

REFERENCES 

 

[1]  A. S. Fotheringham, C. Brundson and M. Charlthon, Geographically Weighted Regression : The Analysis 
of Spatially Varying Relationships, United Kingdom: John Wiley and Sons, Ltd, 2002.  

[2]  C. L. Mei, Geographycally Weighted regression Technique for Spatial Data Analysis, China: Jiaotong 
University, 2005.  

[3]  M. M. F. a. A. Getis, Handbook of Applied Spatial Analysis, Berlin: Heidelberg and New York, 2010, pp. 
255-278. 

[4]  L. Anselin, Spatial Econometrics : Methods and Models, Netherlands: Kluwer Academic Publishers, 
1988.  

[5]  Y. Leung, C. L. Mei and W. X. Zhang, "Statistical Test for Spatial Non-Stationary Based on the 
Geographically Weighted Regression Model," Environment and Planning A, vol. 32, pp. 9-32, 2000.  

 



Estimation of Geographically Weighted Regression Case Study on Wet Land Paddy Productivities in 
Tulungagung Regency 

 

Danang Ariyanto 15 

 


	ABSTRACT
	INTRODUCTION
	METHODS
	Where  ,𝒇-𝒊.=,,𝒆-𝒊.-,𝝈-𝟐..−𝟏, ,𝒆-𝒊. is a vector of OLS residuals,  ,𝝈-𝟐. is the variance based on OLS residual,  T= Trace [ WTW + W2] and  𝒁 is an N by (k+1) matrix of normal standard score (z).

	RESULTS AND DISCUSSION
	CONCLUSION
	REFERENCES