Multivariate Adaptive Regression Splines and Bootstrap Aggregating Multivariate Adaptive Regression Splines of Poverty in Central Java


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(4) (2021), Pages 238-245 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: November 25, 2020 Reviewed: February 19, 2021 Accepted: April 11, 2021 
DOI: http://dx.doi.org/10.18860/ca.v6i4.10871    

Multivariate Adaptive Regression Splines and Bootstrap 
Aggregating Multivariate Adaptive Regression Splines of Poverty 

in Central Java 

Ria Dhea LN Karisma1, Juhari 2, Ramadani A. Rosa3 

1,2,3 Department of Mathematics UIN Maulana Malik Ibrahim Malang 
 

Email: riadhea@uin-malang.ac.id, juhari@uin-malang.ac.id, 
ramadaniauiyanarosa@gmail.com 

ABSTRACT  

Poverty population is one of the serious problems in Indonesia. The percentage of population 
poverty used as a means for a statistical instrument to be guidelines to create standard policies 
and evaluations to reduce poverty. The aims of the research are to determine model population 
poverty using Multivariate Adaptive Regression Spline and Bagging MARS then to understand the 
most influence variable population poverty of Central Java Province in 2018. The result of this 
research is the Bagging MARS model showed better accuracy than the MARS model. Since, GCV in 
the Bagging MARS model is 0,009798721 and GCV in the MARS model is 6,985571. The most 
influence variable population poverty of Central Java Province in 2018 based on MARS model is 
the percentage of the old school expectation rate. Then, the most influentce variable based on 
Bagging MARS model is the number of diarrhea disease. 
 

Keywords: Multivariate Adaptive Regression Splines; Bootstrap Aggregating; Generalized Cross-
Validation; Poverty  

INTRODUCTION  

Poverty has concerned problem in the world even in Indonesia. In Indonesia, which 
is developing country, poverty has been affected in economics that it’s showed level of 
welfare. Therefore, it has become a serious problem that must be resolved. 

The growth of economics is the fundamental factor to reduce poverty. Based on BPS 
data, Indonesia has been able to deal with some economics global problem and succeeded 
in increasing economic growth. Some programs realized such as credit procurement 
programs, agricultural development, equitable development, infrastructure 
improvement, to the procurement program Inpress lagging Village (IDT) to help improve 
the community's living standards. The efforts considered significant because it reduced 
the experiencing of gaps between the rich and the underprivileged people and as an effort 
to realize the strategy of human Quality Development [1]. 

According to BPS (Badan Pusat Statistik) or Indonesian Statistics Institution, level of 
poverty in Indonesia has been reduced in recently. The percentage of privileged people 
reduced up to 0, 58% (in year-on-year) and at 2017 was the lowest poverty level rate. The 

http://dx.doi.org/10.18860/ca.v6i4.10871
mailto:ramadaniauiyanarosa@gmail.com


Multivariate Adaptive Regression Splines and Bootstrap Aggregating Multivariate Adaptive 
Regression Splines of Poverty in Central Java 

Ria Dhea LN Karisma 239 

government succeeded in reducing poverty rate by 1.18 million people from an average. 
The government made a system for implement social protection based on a life cycle 
approach at 2018. However, in some areas poverty rate was slowed in reducing poverty 
[2]. 

MARS were introduced assumption about the relationship between the dependent 
and independent variables to estimate general functions of high dimensional data. 
Bagging MARS is a method that improved performance of in MARS method used bootstrap 
replicating. The past researches Karisma & Sri Harini [3] used MARS to find the 
classification of risk factors of ischemic and hemorrhagic patients by MARS method, 
Kilinc, B et al. [4]  research to find models of metal concentrations to determine soil 
pollution by MARS method, etc.  

MARS model used combination from spline method and recursive partition. Then, 
model in spline regression applied using a set of basis function to achieve q-order spline 
regression and estimated using least squares method. It has knot to find out the continuity 
basis function from one region in regression line to others. Otherwise, Bootstrap 
Aggregating (Bagging) used to minimize squared error value. The aimed of the research 
was the influenced poverty factor using MARS and Bagging MARS then it can be used for 
guidelines standard policies and evaluation to reduce poverty. 

 

METHODS  

Poverty is resident who have an average monthly expenditure per capita below the 
poverty line [5]. Poverty influenced by some factors such as human resources, 
employment, inflation, unemployment, population density, health facilities, income, 
scarcity, transportation, education, business capital [6] .  

MARS method used multivariate nonparametric approaches. It has recursive 
partition formed, high dimensional data, and discontinuity data. Bagging MARS is a 
method that used for improve performance on MARS method with bootstrap replicating. 
MARS developed by Recursive Partitioning Regression (RPR) to estimate sub-region in 
each region continuous model in knots [7]. The advantage MARS is unrequired 
standardization, produced accurate results, used in big data, and used for regression 
analysis and classification simultaneously. Bagging MARS Recursive Partitioning 
Regression (RPR) unable to overcome the discontinuous data in knots. Therefore, the RPR 
algorithm used to estimate and correlate data in subregions [8]. The basis function 
explained the relationship between the dependent and independent variables [9] . The 
regression model used basis functions (BF) as follows: 

𝑦 = 𝛽0 ∑ 𝛽𝑚ℎ𝑚(𝑥)
𝑀
𝑚=1   (1) 

 
Where ℎ𝑚 is a set of basis function, and 𝛽𝑚 is a coefficient of ℎ𝑚  in splines basis function 
defined as: 

ℎ𝑚 = ∏ [𝑆𝑘𝑚(𝑥𝑣(𝑘,𝑚) − 𝑡𝑘𝑚)]
+

𝐾𝑚
𝑘=1   (2) 

 
After modified BF with the RPR model, the MARS model obtained as follows: 

𝑓(𝑥) = 𝑎0 + ∑ 𝑎𝑚
𝑀
𝑚=1 ∏ [𝑆𝑘𝑚(𝑥𝑣(𝑘,𝑚) − 𝑡𝑘𝑚)]+

𝐾𝑚
𝑘=1    (3) 

 
where 𝑎0 is a coefficient, 𝑎𝑚 is a coeefficient function basis-m M is a maximum basis, 𝐾𝑚 is 
an interction degree, 𝑥𝑣(𝑘,𝑚) is label of predictor variables, 𝑡𝑘𝑚 is knot of predictor 

variables 𝑥𝑣(𝑘,𝑚), and 𝑆𝑘𝑚 are variables that take values ± 1 [7]. 



Multivariate Adaptive Regression Splines and Bootstrap Aggregating Multivariate Adaptive 
Regression Splines of Poverty in Central Java 

Ria Dhea LN Karisma 240 

In matrix formed, MARS model defined by (4) 
𝑌 = 𝐵𝑎 + 𝜀 , 𝑌 = (𝑌1, … , 𝑌𝑛)

𝑇, 𝑎 = (𝑎0, … , 𝑎𝑀)
𝑇, 𝜀 = (𝜀0, … , 𝜀𝑛)

𝑇 (4) 

𝐵 =

[
 
 
 
 1

∏ [𝑆1𝑚. (𝑥𝑣(1,𝑚) − 𝑡1𝑚)]
𝐾𝑚
𝑘=1 … ∏ [𝑆𝑀𝑚. (𝑥𝑣(𝑀,𝑚) − 𝑡1𝑚)]

𝐾𝑚
𝑘=1

1 ∏ [𝑆2𝑚. (𝑥𝑣(1,𝑚) − 𝑡1𝑚)]
𝐾𝑚
𝑘=1

⋮

1 ∏ [𝑆𝑛𝑚. (𝑥𝑣(1,𝑚) − 𝑡1𝑚)]
𝐾𝑚
𝑘=1

……
…

∏ [𝑆𝑀𝑚. (𝑥𝑣(𝑀,𝑚) − 𝑡1𝑚)]
𝐾𝑚
𝑘=1

⋮

∏ [𝑆𝑀𝑚. (𝑥𝑣(𝑀,𝑚) − 𝑡1𝑚)]
𝐾𝑚
𝑘=1 ]

 
 
 
 

  (5) 

 
The GCV used to find the best model from MARS method, which used smaller is 

better. It is determined value by trial and error combining the number of basis functions 
(BF), maximum interaction (MI), and minimum observation (MO) [4]. The GCV defined as: 

𝐺𝐶𝑉 =
𝑀𝑆𝐸

[1−
𝐶(�̂�)

𝑛
]
2  (6) 

where 𝑀𝑆𝐸 value defined as 
1

𝑛
∑ [𝑦𝑖 − 𝑓𝑀(𝑥𝑖)]

2𝑛
𝑖=1  , and C(M̂) defined as  

C(M̂) = C(M) + dM (7) 
where, C(M) is matrix trace [B(BTB)−1BT] + 1 that is the number of parameters being fit 
and d represents a cost for each basis function optimization [7]. 

The research used data from Sosial Ekonomi Nasional (Susenas), BPS (Badan Pusat 
Statistik) or Indonesian Statistics Institution for Java Province, and BPS Semarang 
Regional. Total data that used in this research was 350. It used MARS and Bagging MARS 
to analyze, then the steps that employed are divided data into training and testing data. 
Then, MARS method resolved by determined data used MARS method with a combination 
Basis Function (BF), Maximum Interaction (MI), and Minimal Observations (MO)[10]. 
Besides, obtained minimum GCV value to determine the best model in MARS and 
interpreted MARS model.  
Bagging MARS method completed by determined Bagging MARS model using 50 
replications. Then, the best model in Bagging MARS method achieved. The last is 
determined variable that the most influenced of poverty in Central Java Province in 2018. 

RESULTS AND DISCUSSION  

Statistics Descriptive 
The descriptive analysis used to determine characteristic poverty in Central Java at 2018 
(Badan Pusat Statistik, 2019) 
 



Multivariate Adaptive Regression Splines and Bootstrap Aggregating Multivariate Adaptive 
Regression Splines of Poverty in Central Java 

Ria Dhea LN Karisma 241 

 
Figure 1. Descriptive Analysis Poverty Population 

Figure 1 showed the percentage of population poverty in Central Java at 2018. The 
histogram illustrated regency areas and the percentage of poverty population in those 
areas. The highest poverty in those areas was Kabupaten Wonosobo with 17,58%. The 
total of poverty population was almost fifth percent. The percentage of population 
poverty occurred by some factors such as social economic, technology, health care and 
others. Then, the lowest population poverty was Kota Semarang from total population. It 
was under one in twenty percent. 
 
 

Modeling Poverty Population MARS and Bagging MARS Methods  
The MARS model showed in matrix pattern (see Figure 3.2). The matrix plot 

discovered relationship between response variable, which is variable the percentage of 
population poverty (𝑌), and predictor variables, which is the  number of diarrhea disease 
(𝑋1), the number of life expectancy (𝑋2), the percentage of Human Development Index 
(HDI) (𝑋3), the percentage of expenditure per capita by non-food commodities (𝑋4), the 
percentage of open unemployement (𝑋5), the number of infant malnutrition (𝑋6), the 
percentage of family planning and birth control (𝑋7),  the percentage of labor force 
participation rate (𝑋8), the percentage of expectation old school (𝑋9), the number of BPJS 
participants (𝑋10). 

0
2
4
6
8

10
12
14
16
18
20

Percentage of Poverty Population in Central 
Java Province in 2018



Multivariate Adaptive Regression Splines and Bootstrap Aggregating Multivariate Adaptive 
Regression Splines of Poverty in Central Java 

Ria Dhea LN Karisma 242 

 

Figure 2. Matrix Plot  Pattern of Poverty Population  

 
Figure 2 illustrated that indicated unclear and difficult patterns of the relationship 

between variables. Then, in each variable has different characteristics on those areas and 
predictor variable was not able to be explained. In addition, nonparametric method used 
in this research which is MARS and Bagging MARS methods. The best model even in MARS 
and Bagging MARS methods indicated by the GCV.  

The GCV in MARS model was 6.985571 and the R-Sq value was 75,7 %. Then, it was 
five predictor variables that significant and affected population poverty. It was 𝑋1, 𝑋6, 𝑋9, 
𝑋8, 𝑋10 using training data 85% and testing data 15%. The MARS model obtained: 

 

f(x) = 12.8 − 0.000235 ∗ max(0, 𝑋1 − 19574) + 0.0107 ∗ max(0, 249 − 𝑋6) – 0.514 ∗ max(0, 
𝑋8 − 67.5) + 7.35 ∗ max(0, 124 − 𝑋9) − 1.34e − 05 ∗ max(0, 597322 − 𝑋10) 

 
Then, the interpretation of MARS model is 
− 0.000235 ∗ max (0, 𝑋1 − 19574) 
When, the value of 𝑋1 was greater than 19574, for every increased number of diarrhea, it 
increased the percentage of the population poverty at 0.000235 in the Central Java 
Province with an average number of cases of diarrhea less than 19574 people. 
 
0. 0107 ∗ max (0, 249 −𝑋6) 
when, the value of 𝑋6 was smaller than 249, for every increased number of infant 
malnutrition, it increased the percentage of the population poverty at 0.0107 in the 
Central Java province with an average number of infant malnutrition less than 249 people. 
 
−0.514 ∗ max (0, 𝑋8 − 68) 
when, the value of 𝑋8 greater than 68, for every, increased in the percentage of labor force 
participation rate, it decreased the percentage of the population poverty by 0.514 in the 
Central Java province with an average percentage participation rate of a labor force more 
than 68 people. 



Multivariate Adaptive Regression Splines and Bootstrap Aggregating Multivariate Adaptive 
Regression Splines of Poverty in Central Java 

Ria Dhea LN Karisma 243 

 
7. 35 ∗ max (0, 12.4 −𝑋9) 
when, the value of 𝑋9 is smaller than 12.4, for every increased the percentage of old school 
expectancy, it decreased the percentage of the population poverty at 7.35 in the Central 
Java province with an average percentage of the old school expectancy is less than 12.4%. 
 
−1, 34𝑒−05∗ max (0, 597322 −𝑋10) 
When, the value of 𝑋10 was smaller than 597322, for every increased number of 
participants BPJS, it decreased the percentage of the population poverty of 0.0000134 in 
the Central Java province with an average number of participants BPJS less than 597322 
people. 
 

In Bagging MARS method that used 50 times replicate the best model obtained at the 
49th replication using minimum GCV. Then, it was six predictor variables that have 
significant value affected population of poverty. It was 𝑋1, 𝑋4, 𝑋6, 𝑋7, 𝑋8, 𝑋10. The GCV was 
0.009431298 and R-Sq value 0.7955023. The model was: 

 
f̂(x) = 11.17643 – 0.0001232638 ∗ max(0, 13503 − 𝑋1) + 0.0001346581 ∗ max (0, 𝑋1 − 
13503) + 1.637211 ∗ max(0, 48.96 − 𝑋4) – 0.6424541 ∗ max(0, 𝑋4 −48.96) – 0.0250127 ∗ 
max(0, 𝑋6 − 52) + 8.251765e − 05 ∗ max(0, 33664 −𝑋7) – 0.0001611239 ∗ max(0, 𝑋7 − 
33664) – 0.07994066 ∗ max(0, 67.03 𝑋8) − 0,1345248 ∗ max(0,  𝑋8 − 67.03) + 1.335112e 
− 05 ∗ max(0, 𝑋10 −763837) (5) 
 

Table 1. Comparison MARS and Bagging MARS Model 

 Significance variables GCV 

MARS 𝑋1, 𝑋6, 𝑋8, 𝑋9, 𝑋10 6.985571 

Bagging MARS 𝑋1, 𝑋4, 𝑋6, 𝑋7, 𝑋8, 𝑋10 0.009798721 

 
Table 1 showed that the GCV of the Bagging MARS model was 0.009798721. Then, MARS 
model was 6.985571. GCV in the Bagging MARS model indicated a better accuracy than 
the MARS model. Since, Bagging MARS model has GCV minimum than MARS model. 
 
Best Variable In MARS and Bagging MARS Methods 

The population poverty of Central Java using MARS model affected by the number of 
diarrhea disease (𝑋1), the number of infant malnutrition (𝑋6), the percentage of labor 
force participation rate (𝑋8), the percentage of expectation old school (𝑋9), and the 
number of participants BPJS (𝑋10). Table 2 is affected population poverty based on 
importance variables from MARS method.   
 

Table 2. Importance Variables MARS Model 

Variable Importance Variables (%) 

𝑋1 40.9 

𝑋6 22.9 

𝑋8 31.7 

𝑋9 100 

𝑋10 50.8 



Multivariate Adaptive Regression Splines and Bootstrap Aggregating Multivariate Adaptive 
Regression Splines of Poverty in Central Java 

Ria Dhea LN Karisma 244 

Moreover, Bagging MARS affected variable by importance variables that showed in 
Table 3. The variables were the number of diarrhea disease (𝑋1), the percentage of 
expenditure per capita by non-food commodities (𝑋4), the percentage of family planning 
and birth control (𝑋7), the percentage of labor force participation rate (𝑋8), the 
percentage of old school expectancy (𝑋9), and the number of participants BPJS (𝑋10).  

 

Table 3. Importance Variables Bagging MARS Model 

Variable Importance Variables 

𝑋1 95.32921 
𝑋4 0.000000 
𝑋7 60.80385 
𝑋8 0.000000 
𝑋10 0.000000 

 
MARS and Bagging MARS method have distinction in importance variables. In MARS 

method the best level of importance variable was 100% which is the percentage of old 
school expectancy (𝑋9) then in Bagging MARS method was 95.33% which is number of 
cases of diarrhea disease (𝑋1).  

CONCLUSIONS 

Bagging MARS methods obtained better accuracy than the MARS model. The most 
influenced variable population of poverty in Central Java at 2018 using MARS method was 
the percentage of old school expectancy(𝑋9), then the Bagging MARS method is the 
variable number of cases of diarrhea disease(𝑋1). 
 

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