Modeling of Total Fertility Rate (TFR) in East Java Province using

Mixed Semiparametric Regression Spline Truncated and

Kernel Approach

Arip Ramadan

1

, I Nyoman Budiantara

1

and Ismaini Zain

2

1

Institut Teknologi Sepuluh Nopember, Jl. Raya ITS Sukolilo Surabaya, Jawa Timur, Indonesia

2

Department of Statistics, Faculty of Mathematics, Computing and Data Science,

Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia

Keywords: Total Fertility Rate, Mix Estimator, Spline Truncated, Kernel, Semiparametric Regression and Knot Point.

Abstract: The problem of population growth in Indonesia year by year is relatively very high. Based on data from

Indonesia Demographic and Health Survey (SDKI), there is a very high increase of TFR and uncontrollable

increase of population in Indonesia. TFR in East Java is very high in 2012 reached 20%. The behavior of TFR

patterns is associated with the variables that are suspected to affect the Unmet Need, the Age Specific Fertility

Rates (ASFR), the Human Development Index (HDI) and the Infant Mortality Rate (IMR) has a very special

pattern. The relationship pattern between TFR and Unmet Need tends to be linear. While the pattern of

relationship between TFR with ASFR and HDI, in contrast to the Unmet Need that tends to be nonlinear

especially change as the increase or decrease value of these variables. The pattern of relationship between

TFR and IMR appears not to follow a certain pattern. Taking into account this description, this study modeled

using mixed semiparametric regression spline truncated and kernel spline models. The best model is a mixed

semiparametric regression spline truncated and kernel model with a combination of knots that has the smallest

GCV of 0.003964 and the coefficient of determination of 97.04%.

1 INTRODUCTION

Population is the most important thing in sustaining

the development of an area because it is both a subject

and an object of development. As the subject of

population development will play a role in achieving

the achievement of economic and social development

that can affect the increase in social welfare, while as

an object of population development is the party that

gets results from the development of a region.

The Total Fertility Rate (TFR) can affect the

development of the population in the future. In 2012,

there was a national TFR increase from 2.41 in 2008

to 2.6 in 2012. Based on the report, there were only

10 provinces that experienced a decline in their

fertility levels, while the rest were observed to

increase. TFR increases experienced by other

provinces ranged from 31 percent to 63 percent

(BKKBN, 2107).

East Java Province is a province that experienced

a very significant decline in TFR since the

introduction of family planning (KB) policies,

namely the TFR of East Java Province had reached

below 2.1 in 2002. However the 2012 East Java

Province TFR experienced a significant increase

since 2002 which had an effect on increasing the

return of TFR Indonesia. TFR East Java experienced

a TFR increase of more than 20 percent, from 2.1 in

2002 to 2.6 in 2012.

To see the shape of the relationship between TFR

and the variables that influence it can use the

regression method. But sometimes the form of

relationships that occur can vary, such as parametric

or nonparametric. Therefore obtaining a mixed

semiparametric spline truncated and kernel

regression model is important because it includes

parametric and nonparametric.

Ramadan, A., Budiantara, I. and Zain, I.

Modeling of Total Fertility Rate (TFR) in East Java Province using Mixed Semiparametric Regression Spline Truncated and Kernel Approach.

DOI: 10.5220/0008520602710277

In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 271-277

ISBN: 978-989-758-407-7

Copyright

c

 2020 by SCITEPRESS – Science and Technology Publications, Lda. All r ights reserved

271

2 LITERATURE REVIEW

2.1 Mixed Semiparametric Regression

Truncated Spline

Semiparametric regression is a combination of

parametric components and nonparametric

components (Budiantara, 2009). In some cases, it can

be found the relationship between response variables

with one predictor variable is linear, but the

relationship with other predictor variables is

unknown. Variables that have known data patterns or

previous information about their data patterns are

classified in parametric components. While the

unknown variable data pattern is classified on

nonparametric components (Ruppert, Wand, Carrol,

2003).

Given data in pairs (

1i

x

,

2i

x

, ,

pi

x

,

1i

t

,

2i

t

,

qi

t

,

1i

z

,

2i

z

, ,

ri

z

,

i

y

) then the semiparametric

model is formulated as follows:

( )

1 2 1 2 1 2

, , , ,t ,t , ,t ,z ,z , ,z ,

i i i pi i i qi i i ri i

y f x x x



=+

1,2, ,in=

,

with the curve assumed to be additive, it is obtained:

1 1 1

( ) ( ) (z )

pq

r

i ji si ki i

j s k

y f x g t h



= = =

= + + +

  

;

1,2, ,in=

,

where

i

y

is response variable,

( )

1

p

ji

j

fx

=



is a

parametric component,

( )

1

q

si

s

gt

=



is a spline

nonparametric component,

( )

1

r

ki

s

hz

=



are kernel

nonparametric components, and

i



is a random error

that is assumed to be identical, independent and

normally distributed with a zero mean and variant

2



(Eubank, 1999).

2.2 Selection of Knot Point and

Optimal Bandwith

In nonparametric and semiparametric regression with

the Spline approach, the important thing that plays a

role in getting the Spline estimator is the optimal

selection of knot points. Meanwhile the kernel

depends on bandwidth. Bandwidth α is a smoothing

parameter that serves to control the smoothness of the

estimated curve. The knots and bandwidth points that

are too small will produce very rough and fluctuating

curves, whereas the point of the knots or bandwidth

that is too large or wide will produce a very smooth

curve, but not in accordance with the data pattern

(Hardle, 1994). The selection of knot points k and

optimum bandwidth using GCV is defined as follows:

( )

 

( )

2

1

,

MSE k

GCV k

n trace I M k



−

=

−

.

2.3 Residual Assumption Testing

In the semiparametric regression model linear

truncated Spline is assumed to be a random error with

an independent normal distribution with zero mean

and variance

2



(Wahba, 1990). Therefore, before

analyzing and making decisions from the modeling

results, the residual assumption is tested first. The

residual assumption test is an independent test,

identical test and normality test (Tupen, 2011).

3 METHODS

3.1 Overview of General Object

This study uses district / city data in East Java

Province. The East Java Province consists of 38

regions covering 29 regencies and 9 cities. In detail

can be stated in Table 1.

3.2 Research Variable

The response variables used in this study are

categorical data, namely TFR by district / city in East

Java Province in 2015 and variables that are thought

to affect TFR. These variables can be described in

Table 2.

3.3 Research Step

Model TFR using spline truncated semiparametric

regression and kernel. The steps taken are:

1. Plot the response variable with all predictor

variables.

ICMIs 2018 - International Conference on Mathematics and Islam

272

Table 1: List of Regional Names in East Java Province

NO

NAME

REG/CITY

N

O

NAME

REG/CITY

NO

NAME

REG/CITY

1

Pacitan

14

Pasuruan

27

Sampang

2

Ponorogo

15

Sidoarjo

28

Pemekasan

3

Trenggalek

16

Mojokerto

29

Sumenep

4

Tulungagung

17

Jombang

30

Kota Kediri

5

Blitar

18

Nganjuk

31

Kota Blitar

6

Kediri

19

Madiun

32

Kota Malang

7

Malang

20

Magetan

33

Probolinggo

8

Lumajang

21

Ngawi

34

Kota

Pasuruan

9

Jember

22

Bojonegoro

35

Kota

Mojokerto

1

0

Banyuwangi

23

Tuban

36

Kota Madiun

1

Bondowoso

24

Lamongan

37

Kota

Surabaya

1

2

Situbondo

25

Gresik

38

Kota Batu

1

3

Probolinggo

26

Bangkalan

Table 2: Variable Operational Definition

2. Determine parametric component variables and

truncated spline and nonparametric kernels.

3. Model response variables and predictor

variables using spline truncated mix estimators

and kernels in semiparametric regression.

4. Model data with a semiparametric regression of

mixed truncated spline and kernel with one, two,

three, and a combination of knots.

5. Select the optimal point knots, parameters and

bandwidth α based on the GCV method.

6. Testing the assumption of independent, identical

and normal distribution for residuals.

4 RESULT AND DISCUSSION

4.1 Descriptive Analysis

The results of descriptive statistics can be used to

initiate knot points in the next analysis stage.

Table 3: Descriptive Statistics of Response Variables and

Predictor Variables.

Var

Mean

Min

Max

Std Dev

Range

Y

2,06

1,52

2,45

0,21

0,93

1

X

69,11

58,18

80,05

5,40

21,87

1

T

16,01

6,87

31,77

5,01

24,9

2

T

36,21

6,40

87,00

18,79

80,60

1

Z

30,92

17,27

60,51

12,09

43,24

From Table 3 above can be seen the

characteristics of each variable, both the response

variable and predictor variables.

The following is a description of the

characteristics for each predictor variable, namely

Unmet Need

( )

1

X

, ASFR

( )

1

T

, HDI

( )

2

T

and IMR

( )

1

Z

.

a. The average Unmet Need in East Java Province

was 16.01 with a standard deviation of 5.01. The

highest Unmet Need in Bangkalan Regency was

31.77 and the lowest was in Bondowoso Regency

with 6.87 with a range of 24.90.

b. The average ASFR in East Java Province was

36.21 with a standard deviation of 18.79. The

highest ASFR in Bondowoso Regency was 87.00

and the lowest was in Malang City with 6.40.

c. The average HDI in East Java Province was 69.11

with a standard deviation of 5.4. The highest

Human Development Index in Malang City was

Var

Variable

Name

Operational definition

Y

TFR (Total

Fertility

Rate)

The average number of children born

to a woman from the beginning of

childbearing age to the end of her

reproductive period

1

X

Unmet Need

percentage figures that indicate

unmet family planning needs or the

proportion of women of childbearing

age who are married or live together

(sexually active) who do not want to

have more children or who want to

Arrange the next birth within a

minimum period of 2 years but do

not use contraceptive devices or

methods in a district / city

2

X

ASFR (Age

Specific

Fertility

Rate)

The number of births per 1000

women in certain age groups

between 15-49 year.

3

X

HDI (Human

Development

Index)

Measurement of life expectancy,

literacy, education and living

standards for all countries

throughout the world.

4

X

IMR (Infant

Mortality

Rate)

The number of infant deaths in one

particular year per 1000 live births in

the same year

Modeling of Total Fertility Rate (TFR) in East Java Province using Mixed Semiparametric Regression Spline Truncated and Kernel

Approach

273

80.05 and the lowest was in Sampang District

with 58.18 with a range of 24.90.

d. The average IMR in East Java Province was 30.92

or 31 people per 1000 live births with a standard

deviation of 12.09. The highest IMR in

Probolinggo Regency was 60.51 or 61 people and

the lowest was in Blitar City with a number of

17.27 or 17 people with a range of 43.24.

4.2 TFR Modeling using Semipara-

metric Regression Mixture of

Truncated Spline and Kernel

4.2.1 Determination of Parametric

Component and Nonparametric

Component Variables

Figure 1. Scatter Plot Response variables and predictors

A summary of the results of the determination of

parametric components and nonparametric

components is presented in Table 4.

Table 4: Parametric and Nonparametric Components

Notation

Variable

Component

1

X

Unmet Need

Parametric

1

T

ASFR

Nonparametric

2

T

HDI

Nonparametric

1

Z

IMR

Nonparametric

4.2.2 Modeling TFR in East Java Province

Using Semiparametric Mixed

Truncated Splines and Kernel One

Knots

The semiparametric regression model is linear

truncated spline with one parametric component

variable and four nonparametric component variables

with one knot point are as follows:

( )

1

0 1 11 1 21 1 11 12 2

1

22 2 12

y

(z ) ; , ,38.

i i i i i

i i i

x t t K t

t K h i 1,2

    



+

= + + + − + +

− + + =

GCV values generated using semiparametric

regression of truncated spline and kernel mixtures

with one knot are presented in Table 5.

Table 5: Comparison of GCV Values using One Knot Point.

Spline

Kernel

GCV

2

R

Knot

Bandwidth

1 11

t = K

2 21

t = K



56.30

71.72

0.0485

0.004164

95.06

60.13

72.76

0.0494

0.004268

94.69

52.46

70.68

0.0480

0.004301

94.99

63.97

73.80

0.0501

0.004450

94.28

Based on Table 5 the minimum GCV value

produced is equal to 0.004164. Location of knots

points on variables

( )

1

t

that is 56.3

( )

11

K

and

( )

2

t

that is 71.72

( )

12

K

while the bandwidth provided is

as big as

0,0485



=

.

4.2.3 Model TFR in East Java Province with

Truncated Two Point Knot Spline

Components

The truncated spline semiparametric regression

model using two knots with one parametric

component predictor and five nonparametric

components are as follows:

( ) ( )

11

0 1 11 1 21 1 11 31 1 21

11

12 2 22 2 12 32 2 22

y

(z )

, ,38.

i i i i i

x t t K t K

t t K t K h

i 1,2

    

   

++

= + + + − + − +

+ − + − + +

=

The GCV values produced using semiparametric

regression of truncated and kernel spline mixtures

with one knot are presented in Table 6. Based on

353025201510

2.50

2.25

2.00

1.75

1.50

X1 (UN)

Y

Scatterplot of Y vs X1 (UN)

9080706050403020100

2.50

2.25

2.00

1.75

1.50

T1 (ASFR)

Y

Scatterplot of Y vs T1 (ASFR)

8075706560

2.50

2.25

2.00

1.75

1.50

T2 (IPM)

Y

Scatterplot of Y vs T2 (IPM)

6050403020

2.50

2.25

2.00

1.75

1.50

Z1 (AKB)

Y

Scatterplot of Y vs Z1 (AKB)

ICMIs 2018 - International Conference on Mathematics and Islam

274

Table 6 the minimum GCV value produced is equal

to 0.004185. Location of knots points on variables

( )

1

t

that is 68.4

( )

11

K

, 80.8

( )

12

K

, for

( )

2

t

that is

75

( )

21

K

78.37

( )

22

K

while the bandwidth provided

is as big as

0.04472.



=

Table 6: Comparison of GCV Values using Two Knot

Points.

Spline

Kernel

GCV

2

R

Knot

Bandwidth

1 11

t = K

2 21

t = K



1 11

t = K

2 21

t = K

68.4

80.8

0.04472

0.004185

96.32

75

78.37

18.8

56

0.04909

0.004207

95.32

61.54

71.64

68.4

74.6

0.04553

0.004303

95.91

75

76.69

12.6

56

0.05001

0.004346

95.09

59.86

71.64

4.2.4 Modeling TFR in East Java Province

with Truncated Three Point Spots

Components

The truncated spline semiparametric regression

model using three point knots with one parametric

component predictor and five nonparametric

components are as follows:

( ) ( )

( ) ( ) ( )

( )

11

0 1 11 1 21 1 11 31 1 21

1 1 1

41 1 21 12 2 22 2 12 32 2 22

1

42 2 32

y

(z ) ; , ,38.

i i i i i

i i i i

i i i

x t t K t K

t K t t K t K

t K h i 1,2

    

   



++

+ + +

+

= + + + − + − +

− + + − + − +

− + + =

GCV values produced using a semiparametric

regression of truncated spline and kernel mixture with

three knots are presented in Table 7. Based on Table

7, the minimum GCV value produced is equal to

0.004115 Location of the knots on the variable

( )

1

t

that is 57.69

( )

11

K

72.35

( )

12

K

79.67

( )

13

K

for

( )

2

t

that is 72.1

( )

21

K

76.07

( )

22

K

78.06

( )

23

K

while the

bandwidth provided is as big as

0.04277



=

.

4.2.5 TFR model in East Java Province with

a Knot Spots Truncated Combination

Component

The selection of a combination of knots is done by

combining the optimum knots that have been

obtained previously from the calculation of 1 knot, 2

knots and 3 knots. Furthermore, the minimum GCV

will be calculated based on the combination obtained

and the model chosen with the minimum GCV among

the combinations. A truncated spline semiparametric

regression model using a combination of point knots

with a predictor of one parametric component and

three nonparametric components are as follows.

( ) ( )

( )

11

0 1 11 1 21 1 11 31 1 21

11

12 2 22 2 12 32 2 22

1

42 2 32

y

(z ) ; , ,38.

i i i i i

i i i

x t t K t K

t t K t K

t K h i 1,2

    

  



++

+

= + + + − + − +

+ − + − +

− + + =

GCV values produced using semiparametric

regression of truncated and kernel spline mixtures

with a combination of knots are presented in Table 8.

Based on Table 8 the minimum GCV value generated

is equal to 0.003964 with a combination of knot

points, 2.3 and the location of the knot points on the

variable

( )

1

t

that is 68.4

( )

11

K

, 80.8

( )

12

K

and for

( )

2

t

that is 72,1

( )

21

K

76.074

( )

22

K

, 78.062

( )

23

K

while the bandwidth provided is as big as

0.0427



=

.

Table 7: Comparison of GCV Values using Three Point

Knots

Spline

Kernel

GCV

2

R

Knot

Bandwidth

1 11

t = K

1 12

t = K



2 21

t = K

2 22

t = K

3 31

t = K

3 32

t = K

57.69

72.1

72.35

76.07

0.04277

0.004

115

97.0

5

79.67

78.06

65.02

74.09

72.35

76.07

0.04357

0.004

23

96.6

6

79.67

78.06

50.36

70.11

72.35

76.07

0.04242

0.004

295

97.0

5

79.67

78.06

43.04

68.12

72.35

76.07

0.04213

0.004

412

97.1

2

79.67

78.06

Modeling of Total Fertility Rate (TFR) in East Java Province using Mixed Semiparametric Regression Spline Truncated and Kernel

Approach

275

4.2.6 Selection of the best model

Based on the GCV value for each knot point that has

been calculated previously, the next best model is

selected by comparing the GCV values generated by

each model show in Table 9.

Table 8: Comparison of GCV Values using Knot Point

Combinations.

Spline

Kernel

GCV

Com

bina

tion

2

R

Knot

Band

width

1 11

t = K

2 21

t = K



1 12

t = K

2 22

t = K

1 13

t = K

2 23

t = K

68.4

72.1

80.8

76.074

0.0427

0.0039

2.3

97.04

78.062

56.3

72.1

76.074

0.04275

0.0040

1.3

96.98

78.062

68.4

71.72

80.8

0.04848

0.0040

2.1

95.12

57.69

71.72

72.345

0.04865

0.0042

3.1

95.11

79.673

Table 9: Minimum GCV Value on Each Model

Number of Knots

GCV

R-Square

1 Knot Point

0.004164

95.06 %

2 Knot Point

0.004185

96.32 %

3 Knot Point

0.004115

97.05 %

Knot Point Combination

0.003964

97.04 %*

Table 9 shows that the minimum GCV value is

found in the combination of knot points. The model

chosen is the Spline model with three knots. After

obtaining the minimum GCV score for the linear

truncated Spline model, the next step calculates the

estimate for the linear truncated Spline model.

Estimated linear truncated Spline model with a

combination of knot points as follows.

( ) ( )

( )

11

1

22

1

2

0,17519302433 0,00012335311 0,00079110874

0,00166672135 68, 4 0,00055557378 80,80

0,00329040416 0, 01911809653 72,1

0,08784561457 76,07 0,197

ˆ

y

i i i

ii

i

xt

tt

t

++

+

− − −=+

− + − −

−−

−+

−

( )

2

11

2

11

1

2

1

2 0, 0427

1

2 0, 0427

1

11

0,0427

2

y

1

10550155 78

1

0,0427

2

,062

i

zz

n

i

zz

n

i

t

e



+

−

=

−

=

−+









The linear truncated Spline regression model of

the combination of these knots has

2

R

as big as

97.04%. This means that this model can explain TFR

as much 97.04%.

5 CONCLUSION

Applications in TFR data in East Java Province in

2015 got the following results:

1. The best model obtained is by using a

combination of knot points with the following

equation.

( ) ( )

( )

11

1

22

1

2

0,17519302433 0,00012335311 0,00079110874

0,00166672135 68, 4 0,00055557378 80,80

0,00329040416 0, 01911809653 72,1

0,08784561457 76,07 0,197

ˆ

y

i i i

ii

i

xt

tt

t

++

+

− − −=+

− + − −

−−

−+

−

( )

2

11

2

11

1

2

1

2 0,0427

1

2 0,0427

1

11

0,0427

2

y

1

10550155 78

1

0,0427

2

,062

i

zz

n

i

zz

n

i

t

e



+

−

=

−

=

−+









2. The coefficient of determination (

2

R

) the amount

obtained is equal 97.04 percent, so that the model

is suitable for use.

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ICMIs 2018 - International Conference on Mathematics and Islam

276

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Nonparametrik Spline, Tesis, Jurusan Statistika,

Fakultas Matematika dan Ilmu Pengetahuan Alam,

Institut Teknologi Sepuluh Nopember, Surabaya.

Wahba, G., 1990. Spline Models for Observational Data.

Philadelphia: Society.

Modeling of Total Fertility Rate (TFR) in East Java Province using Mixed Semiparametric Regression Spline Truncated and Kernel

Approach

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