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https://doi.org/10.30598/barekengvol18iss2pp1317-1332
June 2024 Volume 18 Issue 2 Page 1317–1332
P-ISSN: 1978-7227 E-ISSN: 2615-3017
BAREKENG: Journal of Mathematics and Its Applications
1317
SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE
POVERTY IN EAST JAVA PROVINCE USING SEM WITH SPATIAL
WEIGHT VARIATIONS
Ashabul Akbar Maulana1, Achmad Fauzan2*
1,2Department of Statistics, Faculty of Mathematica and Natural Sciences, Universitas Islam Indonesia
Jln. Kaliurang No.Km. 14,5, Krawitan, Umbulmartani, Kec. Ngemplak, Kabupaten Sleman, Daerah
Istimewa Yogyakarta 55584, Indonesia
Corresponding author’s e-mail: * achmadfauzan@uii.ac.id
ABSTRACT
Article History:
The East Java Province is one of Indonesia's regions grappling with a notably elevated
poverty rate, accounting for 11.32% of the populace. A strategic approach to comprehending
and redressing this issue involves applying spatial analysis, wherein spatial factors are
intricately integrated into the modeling and cartographic representation of poverty data. The
primary objective of this research is to discern the principal determinants influencing the
incidence of poverty in East Java Province, employing data reflective of the population's
poverty percentages within the province for the year 2021. The study incorporates six pivotal
variables, namely the population poverty rate, open unemployment rate, labor force
participation rate, average years of schooling, adjusted per capita expenditure, and the gross
regional domestic product (GRDP), predicated on adjusted expenditure. Diverse weighting
schemes are applied based on distance and contiguity. The optimal predictive model utilized
is the Spatial Error Model (SEM) incorporating a Distance Band Weighing (DBW)
mechanism with a designated maximum distance () of 75000 meters. Outcomes indicate
that the variable wielding the most substantial influence on the poverty percentage in East
Java Province is the average years of schooling. Specifically, an increase in the pursuit of
formal education manifests as a negative correlation to the poverty percentage, implying an
inverse relationship. Moreover, the SEM model adheres to the requisite assumptions,
encompassing the normality of residuals, homogeneity of residuals, and non-spatial
autocorrelation of residuals. Comparative analyses show that the SEM model using DBW
results in lower MAE, MSE, RMSE, AIC, and MAPE values compared to linear regression.
Additionally, the SEM model has higher pseudo-R² values. Likelihood ratio tests highlight
significant differences, with SEM being more efficient and providing better explanatory power
for dataset variations.
Received: 7th February 2024
Revised: 3rd March 2024
Accepted: 6th May 2024
Published: 1st June 2024
Keywords:
Poverty;
Spatial Error Model;
Spatial Weight Matrix.
This article is an open access article distributed under the terms and conditions of the
Creative Commons Attribution-ShareAlike 4.0 International License.
How to cite this article:
A. A. Maulana and A. Fauzan., “SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE POVERTY IN EAST JAVA PROVINCE
USING SEM WITH SPATIAL WEIGHT VARIATIONS,” BAREKENG: J. Math. & App., vol. 18, iss. 2, pp. 1317-1332, June, 2024.
Copyright © 2024 Author(s)
Journal homepage: https://ojs3.unpatti.ac.id/index.php/barekeng/
Journal e-mail: barekeng.math@yahoo.com; barekeng.journal@mail.unpatti.ac.id
Research Article ∙ Open Access
1318 Maulana, et. al. SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE POVERTY IN EAST…
1. INTRODUCTION
Based on data retrieved from the Central Statistics Agency (BPS), there was an increase in the number
of people living in poverty in the East Java Province in 2021, amounting to 153.63 thousand people. This
number increased by 3.48% from the previous year. The impact of this increase worsened the poverty rate in
East Java Province from 4.42 million people (11.02%) in 2020 to 4.57 million people (11.32%) in 2021, an
increase of 0.30%, which made the province become the third poorest on the entire island of Java. Many
factors can cause poverty, including unemployment, participation, level of formal education, and income [1].
One of the analyses in statistics used for examining and modeling relationships between variables is
regression analysis. The assumptions that must be met in regression analysis include: (1) the relationship
between the dependent variable (y) and the predictor (x) is linear; (2) the error has a mean of zero; (3) the
error has a constant variance; (4) the error is not correlated (autocorrelation or with response); and (5) the
error is normally distributed [2]. But sometimes, these regression assumptions are not met. When related to
data containing location, one method that can be used to analyze this data is spatial data analysis, including
spatial regression, which is caused by spatial dependence, such as the spatial error model (SEM), spatial lag
of X (SLX), spatial autoregressive (SAR), and so on [3], [4]. SAR is widely regarded as the most prevalent
specification and the most universally applicable approach to conceptualizing spatial dependency.
Alternatively, it is feasible to incorporate spatial correlation by including the error factor in the regression
equation. While SAR considers spatial dependency to be of significant importance, SEM regards it as an
unwanted factor. This model only tries to guess the regression parameters for the critical explanatory
variables. It does not look at the possible importance of geographical clustering or spatial autocorrelation,
which could mean more than just attributional dependency. Instead of assuming that a geographical lag affects
the dependent variable, SEM estimates a model that relaxes the traditional regression model assumption that
errors must be independent [5]. Meanwhile, spatial regression models caused by spatial heterogeneity can
use Geographically Weighted Regression (GWR), Geographically Weighted Poisson Regression models, or
other models [6], [7].
Several studies that use spatial analysis, including Sihombing [8], are conducting research on variables
that are factors of poverty, such as income levels, consumption, health, education, and relationships in society
using the SAR model. Tumanggor dan Simamora [9] identified factors that influence the Human
Development Index using the SAR model. Safari [10] used the SEM model to determine the factors that
influence food security in South Sulawesi province. Yulianto and Ayuwida [11] aimed to model the level of
poverty in East Java Province using spatial regression. The data used in this research is the poverty level of
East Java Province in 2015 as the dependent variable, as well as a number of independent variables including
Female Head of Household, Number of out-of-school children aged 7-18 years, Number of disabled
individuals, and six other independent variables. Through this research, using the Spatial Error Model (SEM)
method using the Queen Contiguity weighing matrix, the most influential factors on the poverty rate in East
Java Province in 2015 were determined to be the number of disabled individuals and unprotected drinking
water sources.
Apart from that, the research conducted by Jelita [12] carried out spatial modeling on Gini ratio data
for 2015-2017 as a response variable as well as population size, number of poor people, per capita
expenditure, and the district/city Human Development Index in East Java Province for 2015-2017 as a
predictor variable using K-Nearest Neighbor and Distance Band as a spatial weight matrix. The results of this
research, using the Spatial Error Model (SEM) as a spatial regression model and KNN as a weighing matrix,
found that the number of people living in poverty was the factor that had the most influence on the Gini ratio
of East Java Province in 2015-2017. Other research conducted by Aziah et al. [13] studied the influence of
education, per capita income, and the population living in poverty in East Java Province using panel data
regression. The results of this research show that education and per capita income have a significant, negative
effect on the. Meanwhile, population size has a positive effect on the regency/city poverty in East Java
Province.
Based on these previous research, Muryani [14], Azizi [15], Alam [16], and Widiantari [17], this
research aims to determine the factors that most influence the increase in the percentage of poor people in
East Java Province in 2021. Several socio-economic factors are used, such as the open Unemployment Rate,
labor force participation rate, average years of schooling, product per capita, and GRDP based on East Java
Province expenditure in 2021. Moreover, this research considers different spatial weights when analyzing the
distribution of poverty in East Java Province. Spatial weights can reflect location effects and spatial patterns
BAREKENG: J. Math. & App., vol. 18(2), pp. 1317- 1332, June, 2024. 1319
in the data [18], which can help understand the relationship between socio-economic factors and poverty
levels in the region. This research anticipates that the East Java Provincial government take action to reduce
the percentage of poverty by paying more attention to the factors that have the most significant influence.
2. RESEARCH METHODS
2.1 Data and Source of Data
The data used is secondary data retrieved from the East Java Province Central Statistics Agency (BPS)
Website in 2021 (https://jatim.bps.go.id/). This research data uses six (6) variables, which are: population
poverty Rate (), open unemployment rate (), labor force participation rate (), average years of schooling
(), adjusted per capita expenditure (), gross regional domestic product (GRDP) based on adjusted
expenditure (). The data used is presented in Table 1.
Table 1. Research Data
No
Regency
(%)
(%)
(%)
(Year)
(Million Rupiah)
(Trillion Rupiah)
1
Bangkalan
21.57
8.07
68.66
5.96
8673
17152779
2
Banyuwangi
8.07
5.42
72.32
7.42
12217
55471065
3
Batu
4.09
6.57
73.74
9.31
12887
11471435
4
Blitar
9.65
3.66
70.44
7.5
10757
25700019
5
Bojonegoro
13.27
4.82
71.84
7.38
10221
65839509
6
Bondowoso
14.73
4.46
73.89
5.94
10690
13921654
7
Gresik
12.42
8
69.43
9.56
13280
101318686
8
Jember
10.41
5.44
68.97
6.49
9410
54688719
9
Jombang
10
7.09
70.69
8.55
11394
28553448
10
Kediri
11.64
5.15
69.34
8.08
11127
29361672
11
Bllitar City
7.89
6.61
69.96
10.35
13816
4924572
12
Kediri City
7.75
6.37
67.35
10.15
12359
86485594
13
Madiun City
5.09
8.15
66.87
11.37
16095
10748101
14
Malang City
4.62
9.65
67.59
10.41
16663
53309702
15
Mojokerto City
6.39
6.87
67.09
10.47
13610
4976490
16
Pasuruan City
6.88
6.23
71.66
9.33
13354
5914585
17
Probolinggo City
7.44
6.55
69.71
8.95
12245
8361142
18
Lamongan
13.86
4.9
70.72
8.04
11510
27896543
19
Lumajang
10.05
3.51
66.19
6.67
9203
22623402
20
Madiun
11.91
4.99
67.77
7.82
11658
13372330
21
Magetan
10.66
3.86
73.31
8.36
11833
13417032
22
Malang
10.5
5.4
68.49
7.43
10163
68619103
23
Mojokerto
10.62
5.54
70.47
8.64
12844
60198699
24
Nganjuk
11.85
4.98
64.24
7.78
12172
18640685
25
Ngawi
15.57
4.25
72.88
7.26
11459
13823456
26
Pacitan
15.11
2.04
80.57
7.61
8887
11107402
27
Pamekasan
15.3
3.1
65.88
6.7
8804
11496236
28
Pasuruan
9.7
6.03
69.03
7.41
10297
107630268
29
Ponorogo
10.26
4.38
72.63
7.55
9851
14619969
30
Probolinggo
18.91
4.55
73.24
6.12
10969
23664388
31
Sampang
23.76
3.45
70.19
4.86
8790
13984568
32
Sidoarjo
5.93
10.87
66.47
10.72
14578
141000359
33
Situbondo
12.63
3.68
71.63
6.62
9996
13715834
34
Sumenep
20.51
2.31
75.63
5.92
9000
24161351
35
Surabaya
5.23
9.68
67.3
10.5
17862
407726799
36
Trenggalek
12.14
3.53
72.36
7.56
9743
12959018
37
Tuban
16.31
4.68
73.77
7.18
10380
43984689
38
Tulungagung
7.51
4.91
72.26
8.34
10807
27390424
2.2 Spatial Weight Matrix
The spatial weighting matrix () is a crucial component in spatial analysis, providing a
standardized representation of the relationships between n locations concerning a specified row or constant.
1320 Maulana, et. al. SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE POVERTY IN EAST…
Notably, the diagonal elements of matrix W, denoted as , are set to zero under the assumption that no
spatial unit is contiguous with itself, signifying an absence of influence from a location onto itself [19]. The
determination of matrix W typically involves one of two methods: (1) reliance on the distance between
locations or (2) consideration of contiguity. In the context of spatial analysis, the distance measure ()
between the centroid of location-i with coordinates () and location-j with coordinates () can be
computed using various metrics, including Minkowski, Euclidean, and Manhattan distances. Here, and
represent the latitude and longitude coordinates of location-i, while and denote the corresponding
coordinates for location-j [20]. The general formulation of the spatial weighting matrix, accommodating n
locations, is succinctly expressed through Equation (1) [21]. This matrix serves as a fundamental tool for
comprehending the intricate spatial relationships inherent in the dataset, contributing to the robustness of
spatial analyses within the framework of geographic information systems and statistical modeling.
(1)
The various weighting matrices utilized in this study are presented in Table 2 [22].
Table 2. Variation in Spatial Weighting Matrices
No
Basis of
spatial
weighting
matrix
calculation
Types of
weighting
matrices
Concept
Equation
1.
Based on
contiguity.
Queen
Contiguity
(Contiguity
of sides and
angles)
The weight is designated as 1 for
locations that exhibit both side and
angle adjacency with the observed
location, and as 0 otherwise
2.
Based on
distance.
Inverse
Distance
Weight
(IDW)
The distance serves as a measure of
spatial proximity. As the distance
diminishes, the weight
proportionally increases, signifying
that the weight is inversely related to
the distance.
, where
If ommonly known as the Power
Distance Weight
3.
k-Nearest
Neighbor
(k-NN)
Let denote the distance between
the centroids of location-i and
location-j, where . Subsequently,
these distances, denoted as
, are arranged in
ascending order. Subsequently, the k-
NN locations from location-i are
established, denoted as
, with k taking
values from 1 to n-1.
Subsequently, it is considered
neighbors if the distance between
those locations is among the-k nearest
neighbors.
The k-nearest neighbor locations from
location-i are assigned values according
to the following criteria.
4.
Threshold
Weight/
Distance
Band
Weight
(DBW)
In the threshold weight approach, a
predetermined threshold distance,
denoted as , is established. This
value signifies the maximum distance
for determining spatial dependence
between location-i and location-j.
Locations with distances smaller than
the threshold are considered
neighbors.
BAREKENG: J. Math. & App., vol. 18(2), pp. 1317- 1332, June, 2024. 1321
No
Basis of
spatial
weighting
matrix
calculation
Types of
weighting
matrices
Concept
Equation
5.
Uniform
Weight
Uniform weighing allocates equal
weight to all observed areas, providing
each region with the same weight
value, irrespective of its relationship
to other areas. This approach is
commonly adopted when the observed
area exhibits homogeneity with other
regions [23]. The uniform weighing
matrix is defined as follows."
where is the area around area .
2.3 Testing for Spatial Dependence
The determination of a suitable spatial dependence model for the research data is initiated by a thorough
examination of spatial dependence. Broadly, two (2) primary tests for spatial dependence are employed: (1)
Moran's Index and (2) Lagrange Multiplier Test (LM). Moran's Index is utilized to discern the Spatial Error
Model (SEM), whereas the LM test is deployed to identify models such as Spatial Autoregressive (SAR),
SEM, or Generalized Spatial Model (GSM). In spatial econometrics, the LM test was introduced by Anselin
[24] for the Spatial Autoregressive (SAR) and Generalized Spatial Model (GSM). Subsequently, Burridge
[25] extended the LM test for the Spatial Error Model (SEM). Anselin et al. [26] developed a robust LM test.
Both the LM test and the robust LM test operate under the assumption of normally distributed errors. For the
SEM model, Kelejian and Robinson [27] proposed a computationally straightforward test that circumvents
the need for normality assumptions contingent upon a sufficiently large sample size.
The equations for the Lagrange Multiplier (LM) and robust LM tests for the SAR, SEM, and GSM are
presented in Equations (2)-(6). In the SAR model, the null hypothesis () assumes the absence of spatial
lag dependence (ρ=0), with the alternative hypothesis () proposing the existence of spatial dependence
(ρ≠0). The test statistics for LM in SAR () and robust LM () are articulated in Equations (2)
and (3).
(2)
(3)
where e is the error vector from the regression model, y is the dependent variable vector, W is the spatial
weighting matrix, X is the predictor matrix,
is the estimated regression parameter, tr [.] is the trace of a
matrix, and , and is the maximum likelihood function for the error variance
of the regression model. is a matrix of constants and predictors of size and
is the coefficient vector of regression parameters of size . The test criteria
are that is rejected if
or
. For the SEM model, the hypotheses under
investigation are: (no spatial error dependence) and (there is spatial error dependence).
The test statistics for LM and robust LM for the SEM model are presented in Equations (4) and (5).
(4)
1322 Maulana, et. al. SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE POVERTY IN EAST…
(5)
The testing criteria dictate the rejection of the null hypothesis () if either
or
. In the Generalized Spatial Model (GSM), the hypotheses under scrutiny are as follows:
or (signifying no spatial dependence), and or (indicating spatial dependence
in lag and error terms). The test statistic for LM in GSM is delineated by Equation (6).
(6)
The test criteria are that is rejected if
.
2.4 Spatial Dependency Model
In linear regression models, the assumption that must be met is that observations are independent.
Regression analysis is an analytical approach that characterizes the linear relationship between two or more
variables: the independent variable and the dependent variable. The primary goal of regression analysis is to
estimate the variability in the dependent variable influenced by the independent variable in a given
observation [28]. As outlined by Supanggat [29], the regression model is defined by Equation (7).
(7)
In instances characterized by spatial dependence in observations, it becomes imperative to augment the
regression model with a weight matrix that encapsulates the interdependence among locations. Such
dependencies may manifest in the dependent variable, predictors, errors, or their combinations. The
Generalized Spatial Nested (GNS) model can be formally expressed through Equation (8) [30].
(8)
Assuming follows a normal distribution with a mean (μ) equal to 0 and a variance () equal to ,
where is the identity matrix of size , then is a random variable distributed as , y is a
dependent variable vector of size is the autoregressive coefficient of the lagged dependent variable,
is the spatial weighting matrix for the dependent variable of size , is a matrix
of constants and predictors of size is a vector with elements valued as one of size ,
is the coefficient vector of regression parameters of size is the spatial
weighting matrix for predictors of size , is the matrix of predictors of size is
the autoregressive coefficient vector of size is the spatial weighting matrix for errors of size ,
u is the assumed autocorrelated error vector of size is the autoregressive coefficient of errors, and I
is the identity matrix of size . The determination of the spatial regression model can be observed in
Figure 1.
BAREKENG: J. Math. & App., vol. 18(2), pp. 1317- 1332, June, 2024. 1323
Figure 1. Taxonomy Spatial Dependence Model [30]
Based on Figure 1 and Equation (8) several spatial models that may be employed, such as the SAR, SEM,
SLX, and GSM are presented in Table 3.
Table 3. Spatial Model Variations
No
Model
Model equation
Description
1
Spatial Autoregressive
Model (SAR)
o is assumed to follow a normal distribution, be
stochastically independent, identically centered
around zero with a variance of ragam
().
o SAR is a linear regression model wherein spatial
autocorrelation is present in the dependent variable.
2
Spatial Error Model (SEM)
o SEM is a linear regression model characterized by
spatial autocorrelation in its error term.
o The prediction in SEM involves three components:
(1) the smoothing factor , referred to as the
trend, (2) the spatial factor () or signal, and (3)
the Fit, representing the summation of the trend and
signal [31].
3
Spatial lag of X (SLX)
SLX is a regression model characterized by spatial
autocorrelation among the predictor variables.
4
General Spatial Model
(GSM)/ Spatial
Autoregressive
Confused (SAC)/
Spatial Autoregressive
Moving Average (SARMA)
SAC consists of an autoregressive component in the
dependent variable () and an autoregressive
component in the error term ().
2.5 Testing Assumptions and Goodness of Fit Test
Subsequent to obtaining an appropriate model, the subsequent phase involves scrutinizing the
assumptions inherent in the derived model. Assumptions for spatial models encompass (1) normally
distributed residuals, (2) homogeneity of residual variances, and (3) the absence of spatial autocorrelation in
residuals [32], [33]. The normality of residuals is assessed through the Kolmogorov-Smirnov test [34]. The
homogeneity of residual variances is tested using the Breusch-Pagan test [35], whereas the absence of spatial
autocorrelation in residuals is examined through Moran's index [36].
A goodness-of-fit test for the regression model is essential to assess the model's effectiveness in
predicting the relationship between the independent and dependent variables, both overall and for each
utilized independent variable [37]. The goodness-of-fit test involves employing various measures, namely:
(1) Mean Absolute Error (MAE), (2) Mean Square Error (MSE), (3) Root Mean Square Error (RMSE), (4)
pseudo-, (5) Akaike Information Criterion (AIC), and (6) Mean Absolute Percentage Error (MAPE) [38].
A good model is characterized by lower values of MAE, MSE, RMSE, AIC, and MAPE. This is because a
1324 Maulana, et. al. SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE POVERTY IN EAST…
smaller difference between predictions and actual data indicates greater accuracy in the generated predictions
[39]. Similarly, models with lower AIC values are considered more optimal as they efficiently organize data
explanations with the minimum number of parameters [40]. Meanwhile, a higher pseudo- value elucidates
a better-fitting model [41]. The accuracy of the prediction percentage for forecast error (MAPE) is presented
in Table 4 [42].
Table 4. MAPE Criteria for Model Evaluation
No
MAPE Value
Prediction Accuracy
1
MAPE
Precise prediction
2
10% <
Reliable prediction
3
20% <
Prudent prediction
4
Poor forecasting
Next, to compare whether a spatial model significantly differs from a linear model, the likelihood ratio
(LR) test can be conducted. Additionally, LR can be employed to assess the suitability of the formed model.
The LR test criterion is is rejected if
or p-value , where defines that the alternative
model is not deemed suitable [43].
2.6 Research Flow
The research flowchart is depicted in
Figure 2.
Figure 2. Research Flow Diagram [19]
BAREKENG: J. Math. & App., vol. 18(2), pp. 1317- 1332, June, 2024. 1325
3. RESULTS AND DISCUSSION
3.1 Descriptive Statistics
The primary objective of conducting descriptive analysis is to articulate data in a manner that is both
accessible and engaging for readers, facilitating the retrieval of pertinent information. In this study, a
comprehensive descriptive analysis was undertaken on the Population Poverty Rate data for East Java
Province in 2021. The data was stratified into three distinct categories: Low, Medium, and High. Figure 3
visually illustrates the distribution of poverty levels.
Figure 3. Percentage Poverty in the East Java Province at Regency/ City Level
Figure 3 presents a comparison of the poverty rate by regency/city in East Java Province in 2021. In
2021, the average poverty rate in East Java Province was 11.32%, where Batu City was the area with the
lowest poverty rate in East Java Province at 4.09%, as Batu City is a tourist area. Meanwhile, Sampang
Regency was the area with the highest poverty in East Java Province at 23.76%. The number is associated
with farmers making up the majority of Sampang Regency residents. Farmers in this area have relatively low
incomes, so they are unable to improve the population welfare. The subsequent step involves the examination
of multicollinearity. Multicollinearity testing is carried out through the computation of Variance Inflation
Factor (VIF) values, with the results presented in Table 5.
Table 5. Variance Inflation Factor (VIF) values
Variable
VIF Value
3.31
1.37
4.56
5.76
1.53
According to Table 5, the Variance Inflation Factor (VIF) values for all five predictors are sufficiently
low (less than 10) [44]. Consequently, it can be inferred that there is an absence of multicollinearity among
all employed predictors. Following this, linear regression modeling ensues, accompanied by tests for the
significance of its parameters. A backward stepwise regression elimination test [45] is executed to ensure the
retention of only significant variables. Initially encompassing five (5) variables, subsequent elimination
reveals a sole significant variable, namely the Average Years of Schooling variable (). In line with
Equation (7), the regression model equation is elucidated in Equation (9).
(9)
3.2 Estimation of Spatial Model Parameters
Preceding the computation of the spatial regression model, the formulation of a spatial weighting
matrix takes precedence. In this study, the spatial weighting matrix incorporates varied weights, as delineated
in Table 2. As illustrated in Figure 4 (a), a visual representation of the contiguity chain plot is presented
utilizing the Queen-contiguity approach. Modification of the contiguity criteria was necessary; hence,
1326 Maulana, et. al. SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE POVERTY IN EAST…
Surabaya City and Bangkalan District were deemed contiguous, facilitated by the direct connection through
the Surabaya-Madura Bridge (Suramadu), as elucidated in Figure 4 (b).
(a)
(b)
Figure 4. Chain Plot of Spatial Adjacency between Locations Based on Contiguity (a) Queen Contiguity
Approach, (b) Modified Queen Contiguity Approach
In the computation of the distance-weighted matrix, it is imperative to ensure that coordinates are
represented in a projective coordinate system for computational efficiency. In this study, the Universal
Transverse Mercator (UTM) coordinate system for zone 49S was employed, and the conversion from
geographic coordinates to projective coordinates was facilitated using QGIS software [46]. Subsequently,
the Euclidean distance metric was applied. The distance () between the central point of location-i with
coordinates () and location-j with coordinates () is detailed in Equation (10).
(10)
The variable denotes the latitude coordinate of location-i, while corresponds to the latitude
coordinate of location-j. Similarly, signifies the longitude coordinate of location-i, and represents the
longitude coordinate of location-j. Utilizing Equation (10) in conjunction with Table 2.
The weighting assignment in the k-NN matrix will be performed by first determining the value of ,
then calculating the Euclidean distance. Subsequently, the region with the nearest distances will be assigned
a weight of
for the specified k value. In Figure 5 (a) using as an example. Like k-NN, the weighting
assignment in the radial distance matrix is initially determined by establishing a threshold () as a
reference for weight assignment. Subsequently, Euclidean distances are calculated, and weights are assigned
to areas with distances less than the specified threshold. As an example, in Figure 5 (b) a threshold value of
75000 m is used.
(a)
(b)
Figure 5. Chain Plot of Spatial Adjacency between Locations Based on Distance
(a) k-NN (k=3), (b) DBW (=75000 m)
After obtaining the spatial weighting matrices as depicted in Figure 4 and Figure 5, the subsequent
step involves conducting the Lagrange Multiplier (LM) test using Equations (2) through Equations (6). In
BAREKENG: J. Math. & App., vol. 18(2), pp. 1317- 1332, June, 2024. 1327
the SAR model, the test criteria dictate rejecting the null hypothesis () if
or
, or if the p-value is less than Meanwhile, for SEM, the criteria include rejecting if
or
. A significance level of 5% is applied, resulting in
The LM test
results are detailed in Table 6. Table 6. The LM Test Results
No
Spatial
weighting
matrix
SAR
SEM
GSM
Decisions
Conclusion
p-
val
p-
val
p-
val
p-val
GSM
p-
val
1.
Queen
Contiguity
3.00
0.08
0.05
0.82
6.56
0.01
3.61
0.06
6.61
0.04
Do not reject in
, but reject
in and
GSM
SEM
2.
Modified
Queen
Contiguity
0.71
0.39
2.04
0.15
5.19
0.02
6.52
0.011
7.23
0.03
Do not reject in
, but reject
in
and
GSM
SEM
3.
IDW
0.35
0.55
0.22
0.64
1.53
0.21
1.40
0.24
1.75
0.42
All examinations of
LM do not lead to
the rejection of the
null hypothesis ()
Linear
Regression
4.
k-NN (k=1)
0.33
0.57
0.92
0.34
0.06
0.81
0.65
0.42
0.98
0.61
All examinations of
LM do not lead to
the rejection of the
null hypothesis ()
Linear
Regression
5.
k-NN (k=3)
3.06
0.08
0.15
0.70
10.63
0.00
7.72
0.00
10.78
0.00
Do not reject in
, but reject
in
and
GSM
SEM
6.
k-NN (k=5)
0.58
0.45
4.22
0.04
14.01
0.00
17.65
2.6E-
05
18.23
0.00
Do not reject in
, but reject
in
and
GSM
SEM
7.
DBW
(
)
0.88
0.35
1.14
0.29
7.38
0.01
7.64
0.01
8.52
0.01
Do not reject in
, but reject
in
and
GSM
SEM
8.
DBW
(
)
1.36
0.24
0.56
0.45
7.42
0.01
6.62
0.01
7.98
0.02
Do not reject in
, but reject
in
and
GSM
SEM
9.
DBW
(
)
4.39
0.03
0.12
0.73
15.41
0.00
11.13
0.00
15.53
0.00
Reject in ,
, and
GSM
SEM or
GSM
10.
DBW
(
)
3.68
0.05
0.14
0.70
13.75
0.00
10.22
0.00
13.90
0.00
Do not reject in
, but reject
in
and
GSM
SEM
11.
DBW
(
)
4.93
0.03
0.03
0.85
16.24
0.00
11.34
0.00
16.27
0.00
Reject in ,
, and
GSM
SEM or
GSM
12.
Uniform
Weight
0.51
0.47
0.00
1.00
0.51
0.47
0.00
1.00
0.51
0.77
All examinations of
LM do not lead to the
rejection of the null
hypothesis ()
Linear
regression
According to Table 6, among the various feasible weightings considered, the spatial regression model
was established utilizing 9 specific weightings: Queen Contiguity, Modified Queen Contiguity, -NN (
, and ), and DBW (=55000 m, =60000 m, =65000 m, =70000 m, and
=75000 m). In Table 6, the use of the number of k nearest neighbors is done by trial and error and is
assumed to use The First Law of Geography from Tobler which relates that the closer the object is, the greater
the influence it will have [47]. While the determination of the values was performed iteratively,
commencing with a distance of 55000m. This was necessitated by the minimum threshold required to
generate an invertible weighting matrix. If the distance falls below this threshold, a row in the weighting
matrix contains elements that are all zero or lack neighboring regions, resulting in a matrix with a determinant
of 0. Consequently, the weighting matrix becomes non-invertible, rendering it incapable of solving the
equation [48]. From these 9 weightings, AIC values were computed and are presented in Table 7.
1328 Maulana, et. al. SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE POVERTY IN EAST…
Table 7. Comparison of AIC Values among Various Spatial Regression Models
Evaluations
Models
Queen
Contiguity
Modified
Queen
Contiguity
k-NN
k-NN
DBW
(=
55000
m)
DBW
(=
60000
m)
DBW
(=
65000
m)
DBW
(=
70000
m)
DBW
(= 75000
m)
AIC Value
SEM
184.23
185.50
182.85
181.38
183.32
184.22
181.05
181.63
180.98
GSM
-
-
-
-
-
-
182.60
-
182.85
Based on the findings presented in Table 7, the SEM model incorporating DBW weighting
(=75000 m) emerges as the model with the most favorable AIC value. Subsequent analyses will
therefore focus on the application of SEM spatial regression with DBW weighting (=75000). The
formulation of the SEM model is detailed in Equation (11). The components derived from the SEM are
presented in Table 8.
(11)
Table 8. The Components of the SEM with DBW Weighting (=75000 m)
No
Regency/ City
Fit
Trend
Signal
1
Bangkalan
17.53
16.69
0.85
2
Banyuwangi
11.09
13.12
-2.03
3
Batu
7.91
8.49
-0.58
4
Blitar
12.39
12.92
-0.53
5
Bojonegoro
13.78
13.21
0.56
6
Bondowoso
15.15
16.74
-1.59
7
Gresik
8.28
7.88
0.40
8
Jember
13.92
15.39
-1.47
9
Jombang
10.26
10.35
-0.09
10
Kediri
10.99
11.50
-0.51
11
Bllitar City
5.11
5.95
-0.83
12
Kediri City
5.97
6.44
-0.46
13
Madiun City
3.38
3.45
-0.07
14
Malang City
5.02
5.80
-0.78
15
Mojokerto City
5.78
5.65
0.12
16
Pasuruan City
8.21
8.44
-0.23
17
Probolinggo City
7.97
9.37
-1.40
18
Lamongan
12.44
11.60
0.84
19
Lumajang
13.54
14.95
-1.41
No
Regency/ City
Fit
Trend
Signal
20
Madiun
12.01
12.14
-0.13
21
Magetan
11.08
10.82
0.27
22
Malang
12.08
13.09
-1.01
23
Mojokerto
10.10
10.13
-0.03
24
Nganjuk
12.17
12.24
-0.06
25
Ngawi
13.33
13.51
-0.17
26
Pacitan
12.38
12.65
-0.27
27
Pamekasan
17.67
14.88
2.79
28
Pasuruan
12.74
13.14
-0.40
29
Ponorogo
12.98
12.80
0.18
30
Probolinggo
14.27
16.30
-2.02
31
Sampang
20.78
19.38
1.40
32
Sidoarjo
4.90
5.04
-0.14
33
Situbondo
12.49
15.07
-2.59
34
Sumenep
18.33
16.79
1.55
35
Surabaya
6.07
5.58
0.49
36
Trenggalek
12.63
12.77
-0.14
37
Tuban
14.75
13.70
1.04
38
Tulungagung
10.72
10.87
-0.14
The test statistics for each parameter of Equation (11) are detailed in Table 9. Criteria for parameter
testing ( and ) involve rejecting the null hypothesis () if or if the p-value is less than .
With set at 5%, the critical value is 1.96. The coefficients of the autoregressive error are tested using
the Wald test, with the null hypothesis rejected if the Wald value surpasses
[49].
Table 9. Significance Testing of Each Parameter
Parameters
p-val
Decisions
Conclusions
13.15
< 2.2e-16
Rejected
Significant parameters
-8.96
< 2.2e-16
Rejected
Significant parameters
Wald statistic (15.25)
9.38e-05
Rejected
Significant parameters
According to Equation (11), the variable () demonstrates a negative coefficient of -2.45 concerning
the poverty rate in East Java Province. This implies that an increase in the level of formal education pursued
by residents in the area is associated with a reduction in the poverty rate. The results of this study are in line
with several other studies, including Hofmarcher [50], Brown [51], and Tilak [52]. In essence, a higher
attainment of formal education by the population in East Java Province is correlated with a decreased poverty
percentage in the region. This finding suggests practical implications, such as optimizing compulsory
education for children, particularly up to the high school level, especially in areas identified as high-risk in
Figure 3. Moreover, the mandate for compulsory education could gradually be extended to encompass
BAREKENG: J. Math. & App., vol. 18(2), pp. 1317- 1332, June, 2024. 1329
child [53]. Equation (11) yields an autoregressive error coefficient (λ=0.64), signifying that the error in a
district/city will increase by 0.64 times the average error of its neighboring areas, assuming other variables
remain constant. This finding is followed by an assessment of the SEM model assumptions, as
detailed in
Table 10.
Table 10. Testing the Assumptions of the SEM Model.
No
Testing the
Assumptions of
the SEM Model
Test
statistics
value and p-
value
Test criteria
Decisions
Conclusions
1.
Residual Normality
using the
Kolmogorov-
Smirnov test.
D = 0.127
p-val= 0.123
The null hypothesis () is
rejected if b or if the
p-val is less than α. In this context,
posits that the residuals adhere
to a normal distribution [54].
or
p-val
There is insufficient
evidence to reject ; thus,
it can be concluded that
the residuals in the SEM
model exhibit a normal
distribution.
2.
Homogeneity of
residuals using the
Breusch-Pagan
(BP) test.
BP = 0.832
p-val = 0.361
The null hypothesis is
rejected if
or p-val
posits the assumption of
homogeneity of residual variances
is satisfied [55].
or
p-val
There is insufficient
evidence to reject ; thus,
it can be concluded that
the variance of residuals in
the SEM model is
homogeneous
3.
Non-autocorrelation
of residuals using
Moran's Index.
p-val=0.237
The null hypothesis () is
rejected if or p-val <
α,
: defines the absence of spatial
autocorrelation in the residuals
[31].
or
p-val
There is insufficient
evidence to reject ; thus,
it can be concluded that
there is no spatial
autocorrelation in the
residuals.
Next, the determination of model goodness is continued, as presented in Table 11.
Table 11. Determination of Model Goodness
Model
MAE
MSE
RMSE
Pseudo-
AIC
MAPE (%)
SEM using DBW ( 75000 m)
1.85
5.16
2.27
76.19%
180.98
18.77
Linear Regression
2.17
7.20
2.68
66.81
188.85
21.70
Based on Table 11, the SEM model using DBW yields smaller values for MAE, MSE, RMSE, AIC,
and MAPE compared to the linear regression model. Similarly, the pseudo- value obtained for SEM is
larger than that of the linear regression model. Thus, based on these metrics, the SEM model outperforms
linear regression. According to Table 4, the MAPE value for the SEM model falls within the category of
good forecasting. Furthermore, from the analysis of the likelihood ratio (LR) test, a LR value of
7.614
is obtained, which is greater than
, or <0.05=α. Therefore, it can be stated that
the SEM model with DBW (=75000 m) is more efficient than the linear regression model and provides
a significant improvement in the model's ability to explain variation in the data.
4. CONCLUSION
Based on spatial analysis, among the twelve simulated weightings considering both distance and
contiguity variations, the best model for analyzing the poverty rate in East Java Province is the Spatial Error
Model (SEM) using Distance Band Weight (DBW) with a maximum distance value of 75000 m. After
conducting backward stepwise regression elimination, only one predictor variable is found to be significant
1330 Maulana, et. al. SPATIALLY INFORMED INSIGHTS: MODELING PERCENTAGE POVERTY IN EAST…
out of the initial five, namely, Average Years of Schooling (), with the SEM model equation being
where
. The variable has a negative influence on the
Population Poverty Rate, implying that a higher level of formal education pursued by residents in East Java
Province tends to reduce the poverty percentage in the region. The SEM model yields smaller values for
MAE, MSE, RMSE, AIC, and MAPE compared to the linear regression model. Similarly, the pseudo-
value obtained for SEM is larger, indicating that SEM outperforms linear regression based on these metrics.
The likelihood ratio test also reveals a significant difference between the SEM and linear regression models,
with the SEM model showing superior performance.
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