International Journal Of Scientific & Engineering Research The research paper published by IJSER journal is about Modeling of Age Specific Fertility Rates of Jakarta in Indonesia: A Polynomial Model Approach 1

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Modeling of Age Specific Fertility Rates of Jakarta in Indonesia: A Polynomial Model Approach Author: Md. Rafiqul Islam

Abstract―The purpose of this study is to build mathematical models to age specific fertility rates (ASFRs) and forward cumulative ASFRs for Jakarta, Indonesia. For this, the secondary data of ASFRs have been taken from Muhidin (2005). It is observed that ASFRs and forward cumulative ASFRs follow polynomial models. To examine whether they are valid or not, model validation technique, cross-validity prediction power (CVPP) and F-test are applied to those mathematical models.

Keywords― Age Specific Fertility Rates (ASFRs), Polynomial Model , Cross- Validity Prediction Power (CVPP), t-test, F-test.

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1 INTRODUCTION

n Asian region, mathematical modeling in Population Science especially in Demography has been used in a very limited scale. Mathematical model, in modern era, is a very sophisticated mechanism to express data in mathematical formulation. To investigate the relationships among various demographic phenomena, mathematical models are very

3 METHODOLOGY

In this portion, polynomial model is briefly discussed to understand for the convenience of the readers in the following:

3.1 Polynomial

A mathematical expression of the form
helpful not only for demographers but also for all social science researchers in understanding the process for distinguishing among various important and unimportant variables. Finally, model is very essential tools for not only

y  f (x)  a 0  a1x  a 2 x 2

( a n 0) [4]

i

 a 3x 3

 ...  a n x n

population projections but also for population estimations.
where
a i is the coefficient of x (i =1, 2, 3, ..., n) but a1, a2,...,
Indeed, model is essentially an effort to find out structural
relationships and their dynamic behaviors among the various
components or elements in demographic processes.
an are also constants, positive integer,

a 0 is the constant term and n is the

Traditionally, one can figure out some graphs of the demographic parameters. But, in context of Demography, very few of us know what types of mathematical function are more apt for the parameters.
It was showed that age specific fertility rates (ASFRs) follow
slightly modified biquadratic polynomial model where as forward and backward cumulative ASFRs follow quadratic and cubic polynomial model, respectively in the rural area of Bangladesh [1]. It was also reported that ASFRs follow 3rd degree polynomial model where as forward cumulative ASFRs follow quadratic polynomial model, respectively for
Bangladesh [2].
is called a polynomial of degree n and the symbol x, in this
case, is called an indeterminate. If n=0 then it becomes
constant function. If n=1 then it is called polynomial of degree
1 i.e. simple linear function. If n=2 then it is called polynomial of degree 2 i.e. quadratic polynomial, etc. [5].

3.2 Model Fitting

Using the scattered plot of ASFRs for Jakarta, Indonesia by ages in years (Fig. 1 and Fig. 2), it appears that ASFRs can be fitted by polynomial model with respect to ages. Therefore, an nth degree polynomial model is considered and the form of the model is

n

Therefore, an effort has been made here to find what types of models are more appropriate for the case of Jakarta,

y  a 0

a i

i1

x i  u

[6]
Indonesia. Thus, the main objectives of this study are as
follows:
where, x is age group in years; y is ASFRs; a 0 is the

i

i) to build up mathematical models to ASFRs and forward
constant; a i
is the coefficient of

x (i =1, 2, 3, ..., n) and u is

cumulative ASFRs of Jakarta in Indonesia and
ii) to employ cross-validity prediction power (CVPP) and F-test to these models to check how much these models are valid.

2 SOURCES OF DATA

In the present study, to fulfill the above objectives, the secondary data of ASFRs for Jakarta, Indonesia taken from [3] have been used as raw materials that is presented in Table 1.
the stochastic error term of the model. Here, a suitable n is chosen so that the error sum of square is minimum.
Using the dotted plot of forward cumulative ASFRs for Jakarta, Indonesia by ages (Fig. 3 and Fig. 4), it seems that forward cumulative ASFRs follows an nth degree polynomial model and the form of the model is

Dr. Md. Rafiqul Islam

————————————————

Professor and Ex-Chairman

Dept. of Population Science and Human Resource Development

University of Rajshahi

Rajshahi-6205

Bangladesh.

E-mail: rafique_pops@yahoo.com

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International Journal Of Scientific & Engineering Research The research paper published by IJSER journal is about Modeling of Age Specific Fertility Rates of Jakarta in Indonesia: A Polynomial Model Approach 2

ISSN 2229-5518

y  a 0

n

a i

i1

x i  u

3.7 Area of ASFRs

If f(x) is a bounded (i.e., there exists a positive number k such that |f(x)|≤k for all values of x) single-valued continuous function defined in the interval (a, b), a and b being both finite
where, x is age group in years; y is forward cumulative
ASFRs; a is the constant; a is the coefficient of x i (i =1, 2, 3,
..., n) and u is the error term of the model. In this case, a suitable n is selected so that the error sum of square is
quantities and b>a; then the area is defined as the definite integral of f(x) with respect to x within the limits a and b, is expressed by the following way:

b

minimum.
It is to be mentioned here that these models are fitted using
the software STATISTICA.

Area a

f (x) dx

[15]

3.3 Model Validation

To verify how much these models are stable, the cross validity
Where a is lower limit and b is upper limit. In this case f(x) is
the fitted model of ASFRs or forward cumulative ASFRs and
prediction power (CVPP), 2
, is applied. Here
a=15 years, b=49 years, that is, range of integration is 15 to 49
years. In this section, f(x) is termed as integrand. It is to be

2 1 

(n  1)(n  2)(n  1)

n(n  k  1)(n  k  2)

(1 

R 2 ) ; where, n is the

noted that area has been estimated using the software Maple
9.5.

4 NUMERICAL RESULTS AND DISCUSSION

number of classes, k is the number of regressors in the model
and the cross-validated R is the correlation between observed
and predicted values of the dependent variable [7]. The
The polynomial model is assumed for ASFRs of Jakarta, Indonesia in 1994 and the fitted model is:
y = (-1.00032)+(0.101537)x+(-0.00296)x2+(0.000027)x3

shrinkage of the model is

2 - R2 ; where 2
is cross
(i)
validity prediction power & R2 is the coefficient of determination of the model. Moreover, the stability of R2 of the model is equal to 1-shrinkage. The estimated CVPP, 2 ,
corresponding to their R2 and information of model fitting are shown in Table 2.

3.4 F-test

The F-test is used to the model to verify the overall measure of the significance of the model as well as the significance of R2 . The formula for F-test is given by

R 2

t-stat (-7.08343) (7.028244) (-6.36297) (5.628602)
with coefficient of determination R2 = 0.97793 and

2 =0.873886. In this case, it is known cubic polynomial as n

cv

is 3.
The another polynomial model is assumed for ASFRs for
Jakarta, Indonesia in 1990 and the fitted model is:
y = (-1.13506)+(0.115071)x+(-0.00335)x2+(0.00003)x3
(ii)
t-stat (-7.24857) (7.19076) (-6.51223) (5.76155)
with coefficient of determination R2 is 0.97884 and

2 =0.879086. In this case, it is known as cubic polynomial

F

freedom (d.f.);

(m  1)

(1 R 2 )

( n m)

with (m-1, n-m) degrees of
since n is 3.
And, the polynomial model is assumed for forward
cumulative ASFRs for Jakarta, Indonesia in 1994 and the fitted model is
where m is the number of parameters of the fitted model, n is
the number of cases and R2 is the coefficient of determination
in the model [8].

3.5 Velocity curve

To draw the velocity and elasticity curves we fit the
y=(-0.59651)+(0.041783)x+(-0.00044) x2
(iii)
t-stat (-7.95034) (8.504611) (-5.86797)
giving proportion of variance explained (R2) = 0.99119 and

2

polynomial regression model. The velocity curve is just the

cv

is 0.984897. Here n is 2, so, it is known as quadratic
first derivative of the fitted polynomial regression with respect to age [9] , [10] , [11].
Now, the velocity curve is the first derivative of the fitted polynomial and so we obtain
polynomial.
Another polynomial model is considered for forward
cumulative ASFRs for Jakarta, Indonesia in 1990 and the fitted model is

dy

dx

f ( x)  a1  2a2 x  3a3 x

   nan x

n1 .

y=(-0.67207)+(0.047071)x+(-0.00049) x2
(iv)
t-stat (-7.96027) (8.514592) (-5.87298)
Actually, velocity is the rate of change of y with respect to x.

3.6 Elasticity curve

giving proportion of variance explained (R2) = 0.99122 and

2

The elasticity is estimated using the formula mentioned by

cv

is 0.974914. Here n is 2, so, it is known as quadratic
[12] [13] and [14] as


d log y x dy x

f ( x)

polynomial.
It should be noted here that usual models i. e. Gompertz, Makeham, log-linear, semi-loglinear and logistic were also

d log x

y dx y

applied but seemed to be worse fitted in terms of their

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International Journal Of Scientific & Engineering Research The research paper published by IJSER journal is about Modeling of Age Specific Fertility Rates of Jakarta in Indonesia: A Polynomial Model Approach 3

ISSN 2229-5518


shrinkages. Therefore, the findings of those models were not shown here.
The estimated CVPP, 2
, corresponding to their R2 is shown
in Table 2. From this table it is seen that all the fitted models (i)
- (iv) are highly cross- validated and their shrinkages are
0.104044, 0.099754, 0.006293 and 0.016306, respectively. These
fitted models (i)-(iv) will be stable more than 87%, 87.9%,
98.48% and 97%, respectively. Moreover, from this table, it is shown that the parameters of the fitted models (i) - (iv) are highly statistically significant with more than 97%, 97%, 99% and 99% of variance explained respectively. The stability of R2 of these models are more than 89%, 90%, 99% and 99%, respectively.
The calculated values of F-test for the models (i) - (ii) are 44.31
with (3, 3) d.f. and 46.26 with (3, 3) d.f., respectively where as
the corresponding tabulated values for both cases is 29.5 at 1% level of significance. The calculated values of F-test for the models (iii) - (iv) are 225.015 with (2, 4) d.f. and 225.79 with (2,
4) d.f., respectively where as the corresponding tabulated
values for both cases is only 18.00 at 1% level of significance.
Therefore, from these statistics it is seen that these models and their corresponding R2 are highly statistically significant. Hence, these models are fit well. Predicted and residual values of these fitted models are also presented in Table 1.
The velocity and elasticity curve only for ASFRs in 1994 have
been estimated and shown in Fig. 5 and Fig. 6 respectively. Thereafter, proper definite integral is also employed to these fitted models to find out the area bounded by these fitted curves. The areas of these fitted curves are 1.86, 2.28, 8.42 and
9.70 respectively.

TABLE 1

OBSERVED, PREDICTED AND RESIDUAL VALUES OF ASFRS AND

CUMULATIVE ASFRS FOR JAKARTA, INDONESIA

Fig. 1. Observed and Fitted ASFRs in 1994 for Jakarta, Indonesia . X: Age Group and Y: ASFRs.

Fig. 2. Observed and Fitted ASFRs in 1990 for Jakarta, Indonesia . X: Age Group and Y: ASFRs.

TABLE 2

INFORMATION ON MODEL FITTINGS AND ESTIMATED CVPP OF THE

PREDICTED EQUATIONS OF ASFRS AND ITS FORWARD CUMULATIVE

DISTRIBUTION FOR JAKARTA, INDONESIA

Fig. 3. Observed and Fitted Forward Cumulative ASFRs in 1994 for

Jakarta, Indonesia . X: Age Group and Y: Forward Cumulative ASFRs.

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5 CONCLUSION

Mathematical models of ASFRs for Jakarta, Indonesia and forward cumulative ASFRs are fitted. It is observed that ASFRs follows the 3rd degree polynomial model. On the other hand, forward cumulative ASFRs follows 2nd degree polynomial model. Hope that area might be an alternative approximate measure of fertility.

REFERENCES

Fig. 4. Observed and Fitted Forward Cumulative ASFRs in 1990 for

Jakarta, Indonesia . X: Age Group and Y: Forward Cumulative ASFRs.

Fig. 5. Velocity Curve of ASFRs in 1994 for Jakarta, Indonesia . X: Age Group and Y: Velocity

Fig. 6. Elasticity Curve of ASFRs in 1994 for Jakarta, Indonesia . X: Age Group and Y: Elasticity

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[13] M. Chakravarty, "Microeconomics: Theory

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