Indonesi
a
n
Journa
l
of El
ect
ri
cal Engineer
ing
an
d
Comp
ut
er
Scie
nce
Vo
l.
9
, No
.
3
,
Ma
rch
201
8
,
pp
.
637
~
642
IS
S
N:
25
02
-
4752
,
DOI: 10
.11
591/
ijeecs
.
v9.i
3
.
pp
637
-
642
637
Journ
al h
om
e
page
:
http:
//
ia
es
core.c
om/j
ourn
als/i
ndex.
ph
p/ij
eecs
Tracing
Math
ematical Fu
nction
of Age Spe
cific Fe
rtil
ity Rate in
Penin
sular M
alaysia
No
rs
yel
a Muh
amm
ad
N
oo
r
Mathi
vanan
, P
uz
zi
awat
i A
b Ghani
,
Nor
Az
ura Md.Gh
an
i
Ce
nter fo
r
Stat
ist
ic
al
and
Dec
isi
on
Scie
nces
Stud
ie
s
, F
ac
ulty
o
f
Com
pu
te
r & Ma
them
at
ical Sciences
Un
i
ver
sit
i Te
knol
og
i M
ARA
,
Sela
ngor
,
Ma
la
ysi
a
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
J
un
9
, 201
7
Re
vised
N
ov
2
0
, 2
01
7
Accepte
d
Dec
11
, 201
7
The
siz
e,
str
uc
ture,
a
nd
c
ompo
sit
io
n
of
a
popula
ti
on
a
re
a
ff
ect
ed
by
the
fer
ti
li
ty
rates
at
any
po
i
nt
of
ti
m
e.
M
any
researc
he
r
s
took
the
opport
un
it
y
to
ex
plo
it
the
fe
rtil
it
y
rates
in
obta
inin
g
bette
r
fe
rtil
it
y
patte
rn
s
f
or
t
he
ir
co
un
t
ry.
T
he
curve
for
t
he
age
s
pecific
f
erti
li
ty
rat
e
is
consi
ste
nt,
a
nd
this
featur
e
al
lows
t
he
c
urve
to
be
m
at
ched
with
a
m
at
he
m
at
ic
a
l
m
od
el
.
Thi
s
pap
e
r
ai
m
ed
to
ide
ntif
y
the
best
m
at
he
m
at
ic
a
l
m
od
el
that
fits
the
recent
a
ge
sp
eci
fic
fe
rtil
it
y
rate
in
Peninsular
Ma
la
ysi
a.
This
st
u
dy
fitt
ed
the
f
erti
li
ty
data
of
Peninsular
Ma
la
ysi
a
fr
om
1996
to
2014
to
the
four
m
ath
em
atical
m
od
el
s,
wh
ic
h
wer
e
Hadwige
r,
Gam
m
a,
B
et
a,
and
Go
m
per
tz
m
od
el
s.
Fr
om
the
com
par
isons
of
the
four
m
od
el
s,
it
was
fo
un
d
that
the
best
fitt
ed
m
at
he
m
at
ic
a
l
m
od
el
is
H
ad
wige
r
m
od
el
.
In
relat
io
n
to
the
data
of
early
21
st
ce
ntury
,
there
was
an
i
nclin
at
ion
f
or
t
he
best
fitt
ed
m
at
he
m
at
ic
a
l
m
od
el
fr
om
Had
wi
ger
m
od
el
to
Be
ta
m
od
el
.
Hen
ce
,
th
e
best
m
at
he
m
atical
m
od
el
f
or
each
ye
ar
ca
n
be
us
e
d
t
o
conve
rt
a
fer
ti
li
ty
sched
ule
cl
assifi
ed
in
a
five
-
ye
ar
age
gro
up
into
a
fer
ti
li
ty
sche
du
le
f
or
a
sing
le
-
ye
a
r
of
age
in
Pe
nin
s
ul
ar
Ma
la
ysi
a.
This
m
od
el
al
so
ca
n
be
he
lpf
ul
f
or
po
pu
l
at
ion
pro
j
ect
io
ns
by
us
in
g
li
m
it
ed
and
def
ect
ive
d
at
a.
Ke
yw
or
d
s
:
Be
ta
m
od
el
Fertil
it
y
Gam
m
a
m
od
el
Go
m
per
tz
m
odel
Hadwige
r
m
odel
Cop
yri
gh
t
©
201
8
I
ns
ti
tute of
Ad
v
anced
Eng
ineeri
ng
and S
ci
ence
.
All
righ
ts re
ser
ved
.
Corres
pond
in
g
Aut
h
or
:
Nor Azu
ra M
d.Gh
a
ni
,
Ce
nter fo
r
Stat
ist
ic
al
an
d Dec
isi
on
Scie
nces
Stud
ie
s
Faculty
of Com
pu
te
r
& Mat
hem
atical
Sciences
,
U
niv
e
rsiti
Tekno
l
og
i M
ARA
40450 S
hah A
l
a
m
, S
el
ango
r
Ma
la
ysi
a
Em
a
il
:
azur
a@
t
m
sk
.u
it
m
.ed
u.m
y
1.
INTROD
U
CTION
Fertil
it
y
play
s
an
im
po
rtant
r
ole
in
the
gro
wth
of
a
po
pu
l
at
ion
.
T
he
siz
e
,
struc
ture
,
a
nd
com
po
sit
i
on
of
a
popula
ti
on
are
aff
ect
ed
by
the
birth
rate
at
any
po
int
of
tim
e.
Ther
efore,
resea
rch
e
rs
took
the
oppo
r
tun
it
y
to
ex
plo
it
the
f
erti
li
ty
rates
in
the
effor
t
to
obta
in
the
bette
r
fer
ti
li
ty
patte
rn
s
f
or
t
heir
co
un
t
ry.
Fertil
it
y
can
be
def
i
ned
as
th
e
product
or
ou
tpu
t
of
re
pro
duct
ion
i
ns
te
ad
of
the
a
bili
ty
t
o
ha
ve
chil
dr
e
n
[
1].
The
c
om
m
on
m
easur
e
of
fe
r
ti
lity
rates
incl
ud
e
s
the
cr
ude
birt
h
rate
(CB
R),
ge
ner
al
fert
il
i
ty
rate
(
GFR
),
t
otal
fer
ti
li
ty
rate
(TFR),
gross
r
epro
du
ct
io
n
ra
te
(G
RR
)
,
ne
t
reprod
uction
r
at
e
(N
RR
)
,
a
nd
a
ge
-
s
pecific
fer
ti
li
ty
rate
(
AS
FR
).
Fertil
it
y
rates
can
ei
ther
be
per
i
od
base
d
or
co
hort
based
and
are
ref
e
rred
as
a
ben
c
hm
ark
or
in
dica
tor
to
descr
i
be
the
be
hav
i
our of fe
rtil
it
y i
n
a p
op
ula
ti
on
.
Most
of
the
ti
m
e,
CB
R
is
u
sefu
l
in
deter
m
ining
the
grow
t
h
rate
for
the
popula
ti
on
of
a
co
un
try
.
Howe
ver,
TFR
is
com
m
on
ly
us
e
d
as
a
natio
nal
ind
ic
at
or
f
or
th
e
fam
ily
s
iz
e
in
a
countr
y,
and
i
n
m
any
cases
,
age
-
s
pecific
fe
rtil
it
y
rates
(A
SFRs)
a
re
the
m
os
t
fav
ou
re
d
m
easur
em
en
t
du
e
to
the
i
m
pact
of
wom
en’
s
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
9
,
No.
3
,
Ma
rc
h 201
8
:
637
–
642
638
reprod
uctive
a
ge
t
ow
a
rds
fe
rt
il
ity
beh
a
vio
ur
s.
T
he
baseli
ne
f
or
the
re
producti
ve
a
ge
of
wo
m
en
is
in
th
e
ra
nge
of
15
to
49
ye
ars
old.
This
m
easur
e
is
m
or
e
su
it
able
to
descr
i
be
fer
ti
li
ty
becau
se
it
i
s
m
or
e
gear
e
d
to
tho
s
e
who
wer
e
dire
ct
ly
inv
olv
e
d
in
the
birth
pro
cess
[2
]
.
In
Ma
la
ysi
a,
the
TFR
has
decli
ne
d
qu
it
e
ra
pid
ly
f
ro
m
4.0
childre
n
pe
r
w
om
an
in
1980
t
o
3.0
c
hildr
e
n
per
w
om
an
in
2000,
a
nd
t
he
l
at
est
sta
ti
sti
cs
sh
owe
d
t
hat
in
2014,
the
rate
reac
he
d
the
popula
ti
on
r
eplace
m
e
nt
le
vel
of
2.1
childre
n
[
3].
These
decli
ning
fe
rtil
it
y
rate
s
have
raised
c
oncer
ns
reg
a
r
ding
th
e
po
te
ntial
i
m
plica
ti
on
s
on
the
po
pu
la
ti
on
structu
re.
Be
si
des
that,
Ma
la
ysi
a
is
al
so
faci
ng
po
pu
la
ti
on
a
gein
g
as
t
her
e
are
m
or
e
people
w
it
hin
the
a
ge
ra
ng
e
of
60
ye
ar
s
old
an
d
a
bove
,
rat
her
than
c
hildr
e
n
who
are
y
ounger
tha
n
fi
ve
ye
ars
ol
d.
T
his
sit
uation
co
ul
d
le
ad
to
the
ph
eno
m
enon
of
agein
g
popula
ti
on
. H
e
nce,
the TFR
is
an
im
po
rtant
f
erti
li
ty
m
easur
e
m
ent
in
m
on
it
or
i
ng
t
he
fe
rtil
i
ty
beh
avi
or
s
a
r
ou
nd
the g
l
ob
e
.
Althou
gh
m
os
t
stud
ie
s
ha
ve
pro
vid
e
d
a
naly
sis
on
t
he
TFR
in
unde
rsta
nd
i
ng
fer
ti
li
ty
transiti
on,
thi
s
m
easur
em
ent
i
s
act
ually
bu
il
t
fr
om
the
AS
FRs.
More
ove
r,
the
distri
bu
t
ion
of
the
TF
R
is
sensiti
ve
to
the
tim
ing
e
ff
ect
s.
A
cl
ose
r
l
ook
at
the
ti
m
i
ng
of
birth
s
,
e.g
.
,
m
oth
er’
s
age
is
nee
de
d
to
obta
in
a
m
or
e
com
pr
ehe
ns
ive
view
of
th
e
fe
rtil
it
y
beh
a
vio
r
and
c
hanges
over
ti
m
e.
The
patte
rn
s
of
fer
t
il
ity
acro
ss
dif
fer
e
nt
age
gro
up
s
m
a
y
var
y
al
th
ough
the
sam
e
TFRs
are
pro
du
c
ed.
The
re
is
a
po
s
sibil
it
y
that
dif
fer
e
nt
c
ount
ries
hav
e
sim
il
ar
T
FRs
bu
t
with
diff
e
re
nt
com
po
sit
ion
s
of
ASFR
s
su
ch
as
lo
wer
fe
rtil
it
y
fo
r
youn
ger
age
range
rather
tha
n
ol
de
r
age
ra
ng
e
a
nd
vice
ver
sa
.
Norm
al
l
y,
the
patte
rn
of
fe
rtil
it
y
is
hypo
thes
iz
ed
to
decli
ne
in
the
old
e
r
a
ge
ra
ng
e first,
foll
ow
e
d by the
decli
ne
in
the
yo
unge
st age
range
[
4]
.
Most of
t
he
co
un
t
ries h
a
ve
re
corde
d
the AS
FRs for
se
ven
gro
up
s
of
five
-
ye
ar in
te
rv
al
a
ge
of ASFR
s,
i.e.,
15
-
19,
20
-
24,
25
-
29,
30
-
34,
35
-
39,
40
-
44,
a
nd
45
-
49.
I
n
visu
al
iz
in
g
these
ASFR
s,
the
value
poi
nt
f
or
each
gro
up
of
age
inte
rv
al
is
com
bin
ed
with
one
a
no
t
her
in
a
cr
ude
way.
On
e
of
t
he
al
te
rn
at
ives
to
sm
oo
t
he
n
this
curve
is
by
fitt
ing
a
sp
e
ci
fic
m
at
he
m
a
ti
cal
fu
nctio
n
that
su
it
s
the
AS
FRs
.
This
i
dea
is
possibl
e
and
releva
nt
to
be
execu
te
d
as
t
he
fer
ti
li
ty
patte
rn
for
a
ny
co
untrie
s
is
a
ty
pi
cal
bell
sh
a
ped
with
the
rig
ht
sk
ew
ed
de
viati
on
ov
e
r
the
tim
e.
This
fact
has
m
ade
it
po
ssi
ble
to
c
onve
rt
a
fe
rtil
ity
sche
du
le
cl
as
sifie
d
in
a
five
-
ye
a
r
age
group
i
nto
a
fer
ti
li
ty
sched
ule
for
a
si
ng
l
e
-
ye
ar
of
a
ge.
Be
sides
that,
a
su
it
able
m
at
hem
at
ic
al
fu
nctio
n
ca
n
be
hel
pful
f
or
popula
ti
on
pro
j
e
ct
ions
an
d
va
rio
us
ap
proac
he
s
in
dem
og
ra
ph
ic
est
im
ation
by
us
in
g
lim
i
te
d
an
d
def
ect
ive
data.
The
com
m
on
fer
ti
li
ty
sh
ap
e
i
m
plies
that
fer
ti
li
ty
has
a
zero
value
befor
e
e
nterin
g
the
reprod
uctive
a
ge,
i.e.
,
14
ye
a
rs
ol
d.
T
he
n,
t
he
posit
ive
val
ues
inc
rease
in
the
ra
ng
e
of
15
-
19
ye
ar
s
ol
d,
a
nd
peak
at
the
a
ge
group
of
20
-
29
ye
ars
old
.
T
he
fer
ti
li
ty
declin
es
slo
wly
unti
l
the
value
rea
ches
al
m
os
t
ze
ro
by
the ag
e
of
50 a
nd abo
ve.
A
va
riet
y
of
m
at
hem
atical
fu
nctions
ha
d
be
en
pro
pose
d
by
pr
e
vious
resea
rch
e
rs
in
fitt
in
g
the
A
SFRs
curve
th
r
ough
ou
t
t
he
w
or
l
d.
The
m
at
he
m
at
i
cal
functi
ons
inclu
de
the
Ha
dw
i
ger
f
un
ct
io
n
[
5,
6],
the
Be
ta
an
d
Gam
m
a
fu
ncti
on
s
w
hich
are
si
m
il
ar
to
the
Pears
on
ty
pe
I
an
d
I
II
c
urves
[7
]
,
B
rass
pro
cedure
[8
]
,
G
om
per
tz
curve
[
9]
,
Po
l
ynom
ia
l
m
od
els
[10],
Coale
-
Trussel
l
m
od
el
[11],
an
d
cu
bic
sp
li
ne
[
12
]
.
The
m
os
t
fr
eq
uen
tl
y
us
e
d
m
at
he
m
a
ti
cal
fu
nctio
ns
are
the
Ha
dw
iger,
Gam
m
a,
Be
ta
,
Go
m
per
tz
and
C
oale
-
T
ru
ssell
f
unct
io
ns
[
2].
Howe
ver,
the
Coale
-
Tr
us
sel
fu
nc
ti
on
is
widely
us
e
d
to
fit
the
age
-
sp
eci
fic
to
the
m
arit
al
fer
ti
lity
rate
(A
SM
FR).
T
hi
s
functi
on
des
cribes
the
c
ombinati
on
of
thr
ee
aspects,
i.e.
,
fer
ti
li
ty
,
con
tracepti
on,
a
nd
age
at
m
arr
ia
ge
[
15]
.
These
var
i
ou
s
m
at
he
m
at
ic
a
l
fu
nctio
ns
wer
e
app
li
ed
t
o
di
fferent
AS
FRs
a
ll
aro
un
d
the
worl
d,
wh
e
re
the
bes
t
m
at
he
m
atical
functi
on
that
s
uited
the
A
SF
Rs
fo
r
t
he
pa
rtic
ular
popula
ti
on
was
dep
e
nd
ent
on
the ASFRs
distrib
ution. In s
ho
rt, d
i
ff
e
ren
t
po
pu
la
ti
ons m
ay
h
ave
d
i
ff
e
ren
t
m
at
he
m
at
ic
a
l fu
nctio
ns t
hat f
i
t well
with the
ASFR
s d
at
a.
Pr
e
vious
stu
dy
took
the
init
i
at
ive
to
stu
dy
the
m
at
he
m
atical
fu
nc
ti
on
s
that
best
s
uit
AS
FRs
i
n
Peninsular
Ma
la
ysi
a
fr
om
19
60
unti
l
1995
in
pro
vid
i
ng
a
cl
earer
vie
w
of
the
fer
ti
li
ty
patte
rn
i
n
Peni
ns
ul
a
r
Ma
la
ysi
a
[2
]
.
Seve
ral
m
a
them
at
ic
al
fu
nctio
ns
with
pa
ram
et
ers,
i.e.
,
the
Hadwige
r,
Ga
m
m
a,
Be
ta
,
Go
m
per
tz
and
C
oale
-
T
russel
l
fu
nctio
ns
wer
e
com
par
e
d
in
his
stu
dy.
Even
th
ough
the
stud
y
f
ound
that
the
Ha
dwige
r
functi
on
was
t
he
best
m
at
hem
at
ic
al
fu
nctio
n
that
fit
ASF
Rs
for
t
he
la
st
fifteen
ye
ars,
the
rati
onal
it
y
of
t
he
best
functi
on
ov
e
r
t
h
e
ti
m
e
has
of
te
n
bee
n
dis
pu
te
d
by
dem
og
ra
phers
[
13,
14]
.
The
r
efore,
the
ai
m
of
this
stud
y i
s t
o
ide
nt
ify
the m
at
he
m
at
ic
al
f
un
ct
io
n
that
best
fits
the r
ece
nt
fer
ti
li
ty
d
at
a in P
e
ni
ns
ula
r
Ma
la
ysi
a.
2.
RESEA
R
CH MET
HO
D
The dat
a use
d i
n
the
stu
dy incl
ud
e
d
t
he
TFRs
and ASFR
s in
Peninsular
Mal
ay
sia
f
ro
m
1
99
6
to
20
14.
The
A
SFRs
da
ta
fo
r
a
ge
d
15
ye
ars
to
49
ye
a
rs
are
recor
ded
by
a
sing
le
a
ge
interval
i.e.
15,
16,
...
,
49
or
five
-
ye
ar
age
inter
val
i.e.
15
-
19,
20
-
24,
…
,
45
-
49.
T
his
study
us
ed
the
m
i
d
-
point
of
the
a
ge
inter
val
i.e.
17.5,
22.5,
27.
5,
32.
5,
37.
5,
42.5
a
nd
47.5
f
or
fitt
ing
the
data
with
the
m
a
the
m
at
ic
al
fu
nctions.
Let
assu
m
e
that
represe
nts
the
AS
FR
retrie
ve
d
from
vital
st
at
ist
ic
s.
Me
anw
hile,
represe
nts
the
m
at
hem
at
ic
al
fu
nction
that
will
be
fitt
ed
w
it
h
the
A
SF
Rs
data.
The
n,
the
est
im
ation
of
the
can
be
ob
ta
ine
d
with
the
that
m
ini
m
iz
es
the
functi
on
of
,
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
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E
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c Eng &
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m
p
Sci
IS
S
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02
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4752
Tra
ci
ng Ma
t
he
ma
ti
cal
Fu
nction of
Age
Spec
if
ic
Ferti
lity Rate
…
(
Norsyel
a
M
uhamm
ad
Noo
r
Ma
t
hiva
nan
)
639
5
.
17
2
;
u
u
f
u
f
v
(1)
This
f
un
ct
io
n
so
luti
on
can
be
ob
ta
ine
d
with
the
hel
p
of
s
ta
ti
sti
ca
l
pack
a
ges
s
uch
as
S
PSS,
S
-
Pl
us
,
STA
T
IS
T
ICA
,
SA
S
a
nd
R
-
p
rog
ram
m
ing
throu
gh
t
he
nonlinear
e
s
tim
a
ti
on
m
e
tho
ds.
T
he
non
-
li
near
est
i
m
ation
m
od
el
has
bee
n
w
idely
us
ed
in
va
rio
us
fiel
d
of
stud
ie
s
i
nclu
de
in
stu
dy
of
popu
la
ti
on
grow
t
h
[
1],
pr
e
dicti
ve
co
nt
ro
l
[
15,
16]
,
pa
tt
ern
rec
ogniti
on
[17]
et
c.
I
n
this
stud
y,
t
he
p
aram
et
er
was
est
i
m
at
ed
by
t
he
R
-
pro
gr
am
m
ing
s
of
t
war
e.
T
his
s
tud
y
us
e
d
the
c
oeffici
ent
of
de
te
rm
inati
on
,
R
2
to
identify
the
best
m
at
hem
at
ic
a
l
functi
ons
that
f
it
well
with
the
ob
se
rv
e
d
A
SF
R
data.
The
m
a
them
a
ti
cal
fu
nction
f
or
the
A
SFRs
co
ns
ist
of
two
i
m
po
rta
nt
pa
rts su
c
h
as
,
2
(
:
)
(
:
,
.
.
.
,
)
r
f
u
h
u
(
2)
wh
e
re
u
repres
ents
the
age
of
the
m
oth
er
in
a
ll
m
od
el
s
that
are
us
ed
in
t
he
stud
y.
The
n,
)
,....,
:
(
2
r
u
h
θ
θ
is
the
pro
bab
il
it
y
den
sit
y
functi
on
with
)
1
(
r
as
the
par
am
et
er
of
r
θ
θ
,.....,
2
and
t
his
functi
on
are
re
presen
te
d
diff
e
re
ntly
accor
ding
to
the
f
our
m
at
he
m
ati
cal
fu
nctio
ns
i.
e.
Go
m
per
tz
,
Hadwige
r,
Be
t
a
and
Gam
m
a
m
od
el
s.
Me
anwhil
e, th
e
1
is
the
p
a
ram
et
er th
at
repres
ents the
TFR.
2.1. Go
mpertz
Model
The
G
om
per
tz
functi
on
[
9]
is
a
sig
m
oid
f
un
ct
io
n.
It
is
a
sp
eci
al
case
of
the
ge
ner
a
li
sed
log
ist
ic
functi
on.
T
he
double
ex
pone
ntial
functi
on
will
beco
m
e
the
Go
m
per
tz
f
unct
ion
w
hen
t
he
par
am
et
er
of
α
=
λ
.
Me
anwhil
e,
th
e
G
om
per
tz
f
unct
ion
will
bec
om
e
the
G
um
bel
functi
on
if
t
he
β
=
1.
The
G
om
per
tz
f
un
ct
i
on
ca
n
be
e
xpresse
d b
y,
14
w
h
e
r
e
e
x
p
e
x
p
)
(
m
m
u
m
u
u
h
(3)
The
m
val
ue
can
be
de
fine
d
as
the
lowest
m
a
rr
ia
ge
a
ge
of
the
popu
la
ti
on
w
her
e
f
or
this
f
unct
ion,
the
m
was 14.
2.2. Ha
dwi
ger
Model
The Ha
dw
i
ger
functi
on [5] i
s
expresse
d by,
m
u
m
u
m
u
m
u
u
h
w
h
e
r
e
2
e
x
p
)
(
2
2
3
(4)
u
-
m
is
the
a
ge
of
the
m
oth
e
r
at
birth
a
nd
m
is
a
const
ant
and
norm
al
l
y
equ
al
s
to
ze
ro.
The
pa
ram
et
ers
of
the
m
od
el
m
a
y
hav
e
a
dem
ographic
inter
pr
et
at
ion
as
t
he
pa
ra
m
et
er
is
associat
ed
with
total
fer
ti
li
ty
,
α
determ
ines
the
heig
ht
of
t
he
cu
rv
e
a
nd
β
is
relat
ed
to
the
m
ean
age
of
m
oth
erh
oo
d,
w
hile
th
e
te
r
m
is
associat
ed
with
the
m
axi
m
u
m
A
SF
R [17].
2.3. Ga
mma
Model
The Gam
m
a fu
nction [
7] is
giv
en
b
y,
m
u
m
u
m
u
r
u
h
w
h
e
r
e
e
x
p
)
(
)
(
1
)
(
1
(5)
The
m
val
ue
is
the
youn
gest
age
of
the
m
othe
r
to
gi
ve
bi
rth
,
w
hich
was
pr
evio
us
ly
14.
T
he
pa
ram
et
ers
α
a
nd
β
a
re
r
el
at
ed
to
the
m
od
e,
m
ean,
a
nd
var
ia
nce
of
the
f
un
ct
io
n
but
not
in
a
sim
ple
li
near
way
an
d
thus
,
they
do not
ha
ve direct
dem
og
ra
phic
inter
pret
at
ion
s [7].
2.4.
Be
ta M
odel
The
Be
ta
funct
ion
[
7] is gi
ven b
y,
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IS
S
N
:
2502
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4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
9
,
No.
3
,
Ma
rc
h 201
8
:
637
–
642
640
b
u
a
a
b
a
b
a
u
a
b
a
u
B
u
h
w
h
e
r
e
1
)
,
(
)
(
1
1
1
(6)
The
value
f
or
a
re
presents
t
he
youn
gest
age
of
the
m
oth
er
to
gi
ve
bi
rth
w
hich
is
15,
m
e
anwhil
e,
b
re
present
s
the
ol
dest
a
ge
of
t
he
m
oth
er
t
o
giv
e
birth
w
hich
is
50.
The
par
am
et
ers,
α
and
β
re
presen
t
the
lo
wer
a
nd
uppe
r
age lim
i
ts of
fe
rtil
it
y bu
t sh
owed
that i
n
se
ve
r
al
c
ases the
val
ue of
β
fa
r
e
xc
eeds t
he
m
axim
u
m
ag
e.
3.
RESU
LT
S
A
ND AN
ALYSIS
Table
1
s
hows
par
am
et
er
θ
wh
e
re
re
pr
e
se
nts
the
est
im
a
t
ed
TFR
ba
sed
on
the
fou
r
m
at
he
m
at
ic
a
l
m
od
el
s
with
th
e
obser
ve
d
T
F
R.
It
is
fou
nd
t
hat
al
l
m
at
he
m
at
ic
al
m
od
el
s
able
to
est
i
m
at
e
cl
os
el
y
the
val
ue
of
TFR
thr
ough
t
he
ye
ars
.
Wh
e
r
eas
pa
ram
et
ers
α
a
nd
β
do
no
t
hav
e
any
de
m
og
raphic
inte
rpretat
ion
s
exc
ept
f
or
par
am
et
er α o
f
the H
a
dwige
r m
od
el
w
hich
is
r
el
at
ed
t
o
the
m
ean ag
e
of m
oth
e
rho
od.
Table
1.
Para
m
et
er Esti
m
ati
on for Eac
h of
the Mat
hem
at
i
cal
Functi
ons fro
m
1
99
6
-
2014
Year
Ob
serv
ed
TFR
Mathe
m
a
tical f
u
n
ctio
n
of
ASFRs
Go
m
p
e
rtz
Had
wig
er
Ga
m
m
a
Beta
1996
3
.22
3
.34
9
3
.28
5
3
.27
0
3
.19
7
1997
3
.14
3
.26
4
3
.20
2
3
.18
6
3
.11
7
1998
3
.02
3
.15
1
3
.08
9
3
.07
6
3
.00
5
1999
2
.94
3
.05
7
2
.99
6
2
.98
1
2
.91
5
2000
3
.01
3
.11
8
3
.05
6
3
.04
2
2
.97
7
2001
2
.76
2
.86
2
2
.80
5
2
.79
1
2
.73
1
2002
2
.67
2
.76
2
2
.70
6
2
.69
4
2
.63
6
2003
2
.56
2
.64
0
2
.58
6
2
.57
4
2
.52
0
2004
2
.51
2
.58
5
2
.53
1
2
.52
0
2
.46
7
2005
2
.41
2
.48
3
2
.42
8
2
.41
9
2
.36
7
2006
2
.35
2
.42
5
2
.36
9
2
.36
0
2
.30
9
2007
2
.33
2
.40
8
2
.35
0
2
.34
2
2
.29
2
2008
2
.33
2
.40
4
2
.34
2
2
.33
4
2
.28
4
2009
2
.30
2
.37
0
2
.30
8
2
.29
9
2
.25
2
2010
2
.18
2
.24
8
2
.18
9
2
.18
1
2
.13
6
2011
2
.23
2
.29
8
2
.23
4
2
.22
8
2
.18
3
2012
2
.25
2
.31
7
2
.25
3
2
.24
6
2
.20
2
2013
2
.09
2
.14
9
2
.09
4
2
.08
7
2
.04
6
2014
2
.10
2
.17
2
.11
3
2
.10
7
2
.06
6
Ba
sed
on
Tabl
e
2,
Ha
dw
i
ger
m
od
el
was
the
best
m
at
he
m
a
ti
cal
m
od
el
that
fits
AS
FRs
with
99
.
46%
m
at
ched
the
obser
ve
d
data.
Un
li
ke
i
n
19
96
to
2007
w
he
re
the
m
od
el
had
sh
ow
n
the
bes
t
per
f
or
m
ance
a
m
ong
the
f
our
m
at
hem
at
ic
al
m
od
els,
Ha
dwiger
a
nd
Be
ta
m
od
el
s
ha
d
a
sim
i
la
r
R
2
valu
e
in
2008.
T
he
n,
t
he
Be
t
a
m
od
el
see
m
ed
to
fit
the
data
from
20
09
t
o
20
14
bette
r
than
t
he
ot
her
m
od
el
s.
T
he
perform
ance
of
the
Hadwige
r
m
od
el
sli
gh
tl
y
decr
eased
ove
r
the
tim
e.
It
see
m
e
d
that
there
wa
s
an
incli
na
ti
on
of
a
sh
ift
f
rom
the
Hadwige
r
to
the
Be
ta
m
od
el
.
The
thir
d
m
at
hem
atical
mo
del
that
fits
well
with
the
data
was
the
Gam
m
a
m
od
el
.
The
Go
m
per
tz
m
od
el
ha
d
the
l
ow
est
R
2
values
wh
ic
h
in
dicat
e
it
was
the
w
orst
am
on
g
the
m
at
he
m
at
ic
a
l m
od
el
s u
sed
in
the st
ud
y.
The
incli
natio
n
of
a
s
hift
obse
rv
e
d
f
ro
m
Hadwige
r
m
od
el
to
Be
ta
m
od
el
i
s
du
e
t
o
the
c
ha
ng
e
i
n
the
structu
re
of
fe
rtil
it
y
based
on
the
wo
m
en’
s
age
in
t
he
21
st
centur
y.
Thi
s
interest
in
g
fi
nd
i
ng
m
at
ched
tho
se
ob
s
er
ved
i
n
th
e
earli
er
stud
y
wh
e
re
the
Be
ta
m
od
e
l
pro
vide
d
the
be
st
fit
for
m
os
t
of
the
AS
FRs
in
c
ountries
with
the
non
-
enh
a
nce
d
earl
y
-
age
fe
rtil
it
y
[1
7].
T
he
po
pu
la
ti
on
in
Ma
la
ysi
a
al
so
e
xp
e
riences
the
sa
m
e
sit
uation.
The
co
un
try
e
xpe
riences
l
ow
fe
rtil
it
y
rates
for
w
om
en
in
t
heir
ea
rly
reprod
uctive
a
ge
s
.
This
sit
uation
i
nd
ic
at
es
that
w
om
e
n
ei
the
r
delay
ed
thei
r
m
arr
ia
ge
or
pract
ic
e
fa
m
ily
plann
i
ng.
The
f
or
m
ulatio
n
of
the
Be
ta
m
od
e
l
with
inclusi
on
of
a
ge
lim
it
a
ti
on
at
m
arr
ia
ge
su
it
ed
with
the
cha
nges
in
the
cu
rr
e
nt
fe
r
ti
lity
beh
a
viou
rs
highly
relat
ed
t
o
t
he
postp
on
em
ent
of
m
arr
ia
ge
a
m
on
g
w
om
en
in
Pe
ninsular
Ma
la
ysi
a.
Hence
,
it
was
pro
ven
th
at
there
is
a
hi
gh
possi
bili
ty
t
hat
the
Be
ta
m
od
el
will
be
t
he
best
m
at
he
m
at
ic
al
m
od
el
th
at
fits
the ASFR i
n
P
enins
ular
Ma
la
ysi
a com
par
ed t
o
ot
her thre
e
m
at
he
m
at
ic
a
l m
od
el
s in
the
f
uture.
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci
IS
S
N:
25
02
-
4752
Tra
ci
ng Ma
t
he
ma
ti
cal
Fu
nction of
Age
Spec
if
ic
Ferti
lity Rate
…
(
Norsyel
a
M
uhamm
ad
Noo
r
Ma
t
hiva
nan
)
641
Table
2.
C
oeffi
ci
ent of
Deter
m
inati
on
, R
2
for
Eac
h of t
he M
at
hem
a
ti
cal
Functi
ons fr
om 1
996
-
20
14
Year
Mathe
m
a
tical f
u
n
ctio
n
of
ASFRs
Go
m
p
e
rtz
Had
wig
er
Ga
m
m
a
Beta
1996
0
.98
6
1
0
.99
6
0
0
.99
2
0
0
.99
5
3
1997
0
.98
6
0
0
.99
6
0
0
.99
2
2
0
.99
5
3
1998
0
.98
4
6
0
.99
7
4
0
.99
1
1
0
.99
5
4
1999
0
.98
4
8
0
.99
5
5
0
.99
1
9
0
.99
3
3
2000
0
.98
5
1
0
.99
5
4
0
.99
2
6
0
.99
1
5
2001
0
.98
4
0
0
.99
4
5
0
.99
1
3
0
.99
1
0
2002
0
.98
5
3
0
.99
5
9
0
.99
3
0
0
.99
0
7
2003
0
.98
4
4
0
.99
5
4
0
.99
2
5
0
.99
0
5
2004
0
.98
4
9
0
.99
5
8
0
.99
3
1
0
.98
9
7
2005
0
.98
3
7
0
.99
6
2
0
.99
3
1
0
.99
1
5
2006
0
.98
3
4
0
.99
6
6
0
.99
3
4
0
.99
2
1
2007
0
.98
0
9
0
.99
6
1
0
.99
2
4
0
.99
3
2
2008
0
.97
6
5
0
.99
4
9
0
.99
0
1
0
.99
4
9
2009
0
.97
3
9
0
.99
3
6
0
.98
8
4
0
.99
5
0
2010
0
.97
3
2
0
.99
3
3
0
.98
8
1
0
.99
4
4
2011
0
.97
1
3
0
.99
2
5
0
.98
7
4
0
.99
4
5
2012
0
.97
0
5
0
.99
1
9
0
.98
7
0
0
.99
4
2
2013
0
.97
4
2
0
.99
2
1
0
.98
8
4
0
.99
8
1
2014
0
.97
1
5
0
.99
1
0
0
.98
7
0
0
.99
2
9
Av
erage of
R
2
0
.98
0
2
0
.99
4
6
0
.99
0
8
0
.99
3
4.
CONCL
US
I
O
N
The
AS
FR
in
Peninsular
Ma
l
ay
sia
has
a
ty
pical
bell
sh
ape
ov
e
r
ti
m
e
and
it
is
po
ssible
t
o
m
at
ch
the
AS
FR
c
urves
with
s
pecific
m
at
he
m
at
ic
a
l
fu
nctio
ns.
F
our
com
m
on
m
at
he
m
at
ic
al
fu
nctions
i.e.
the
Ha
dw
i
ger,
Gam
m
a,
Be
ta
and
G
om
per
tz
m
od
el
s
are
us
e
d
to
trace
the
best
m
at
he
m
a
t
ic
al
fu
nctio
n
t
hat
fits
the
A
S
FRs
of
ov
e
rall
Pe
nin
s
ular
Ma
la
ysi
a
popula
ti
on
.
T
he
H
a
dw
i
ger
m
od
el
fits t
he be
st from
1
996
t
o 2
007. H
oweve
r,
the
re
was
an
incli
nat
ion
of
a
sh
ift
be
tween
the
Ha
dw
i
ger
a
nd
Be
ta
m
od
el
s
wh
e
re
the
la
tt
er
m
od
el
fits
the
be
st
fr
om
2009
to
2014.
Hen
ce
,
t
her
e
was
a
hi
gh
po
ssibil
it
y
that
the
Be
ta
m
od
el
will
be
t
he
bes
t
m
at
he
m
at
ic
a
l
m
od
e
l
that
fits
the
AS
FRs
in
Peni
nsula
r
Ma
la
ysi
a
in
the
21
st
ce
ntury
.
Eve
n
th
ough
this
stu
dy
pr
ove
d
that
the
f
our
com
m
on
m
at
he
m
at
ic
al
fu
nctions
f
it
well
wi
th
the
data,
oth
er
resea
rch
e
rs
can
ta
ke
op
portu
niti
es
to
us
e
oth
e
r
distrib
utions e.
g.
c
ubic
s
pline,
poly
no
m
ia
l
m
od
el
s
etc. that
m
ay
also f
it
th
e d
at
a.
REFERE
NCE
S
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a
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ase
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